Spike dynamics in primate lateral prefrontal cortex during working memory and decision-making: A fractal analysis.

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Learn more: PMC Disclaimer | PMC Copyright Notice iScience . 2026 Mar 20;29(5):115429. doi: 10.1016/j.isci.2026.115429 Search in PMC Search in PubMed View in NLM Catalog Add to search Spike dynamics in primate lateral prefrontal cortex during working memory and decision-making: A fractal analysis Nazanin Zandi-Mehran Nazanin Zandi-Mehran 1 Neuroscience & Neuroengineering Research Lab., Biomedical Engineering Department, School of Electrical Engineering, Iran University of Science & Technology (IUST), Tehran, Iran Find articles by Nazanin Zandi-Mehran 1 , Ardalan Faezmehr Ardalan Faezmehr 1 Neuroscience & Neuroengineering Research Lab., Biomedical Engineering Department, School of Electrical Engineering, Iran University of Science & Technology (IUST), Tehran, Iran Find articles by Ardalan Faezmehr 1 , Mohammad Reza Daliri Mohammad Reza Daliri 1 Neuroscience & Neuroengineering Research Lab., Biomedical Engineering Department, School of Electrical Engineering, Iran University of Science & Technology (IUST), Tehran, Iran Find articles by Mohammad Reza Daliri 1, 2, ∗ Author information Article notes Copyright and License information 1 Neuroscience & Neuroengineering Research Lab., Biomedical Engineering Department, School of Electrical Engineering, Iran University of Science & Technology (IUST), Tehran, Iran ∗ Corresponding author [email protected] 2 Lead contact Received 2025 Jun 2; Revised 2025 Oct 11; Accepted 2026 Mar 18; Collection date 2026 May 15. © 2026 The Author(s) This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). PMC Copyright notice PMCID: PMC13091740  PMID: 42011177 Summary The lateral prefrontal cortex plays a key role in working memory and decision-making. Using multiunit spike recordings from macaque dorsolateral and ventrolateral prefrontal cortex during a match-to-sample task, we quantified the complexity of inter-spike intervals using Higuchi, Katz, and box-counting fractal dimensions. Our results indicate that fractal dimensions differentiate cognitive processes associated with working memory and decision-making, revealing significant differences in firing patterns across prefrontal regions during distinct time intervals, training phases, and tasks. Functional distinctions between dorsolateral and ventrolateral prefrontal cortex were strongest before training, reflecting preferential involvement in spatial and shape processing. Ultimately, the findings highlight the potential of fractal analysis as a biomarker for cognitive functions and demonstrate the impact of training on the functional dynamics of these brain regions. This study contributes to a deeper understanding of how the lateral prefrontal cortex process spatial and visual characteristics, paving the way for future research in cognitive neuroscience. Subject areas: Biocomputational method, Cognitive neuroscience, Data analysis, Neuroscience, Sensory neuroscience Graphical abstract Open in a new tab Highlights • Fractal analysis reveals changes in complexity of lateral prefrontal spike timing • Complexity changes reflect lateral prefrontal cortex spatial/object recognition roles • Training alters complexity in the lateral prefrontal cortex • Simulation reveals distinct sensitivities of fractal methods Biocomputational method; Cognitive neuroscience; Data analysis; Neuroscience; Sensory neuroscience Introduction Understanding the mechanisms of the human brain is essential for enhancing our comprehension of cognitive processes and mental health, particularly the prefrontal cortex (PFC), which constitutes the highest level of the cortical hierarchy dedicated to the representation and execution of actions. The lateral region of the PFC (LPFC) is a highly developed area in human brain that provides essential cognitive support for the temporal organization of behavior, speech, and reasoning. 1 The role of this area in temporal organization is supported by several interrelated subordinate functions, such as temporal integration, working memory, and cognitive set. 1 Working memory is one of the brain’s fundamental cognitive functions among them and is crucial in facilitating various cognitive processes such as planning, comprehension, decision-making, reasoning, and problem-solving. 2 , 3 , 4 , 5 , 6 To understand the neural basis and circuitry of working memory, many researchers have focused on the functionality of the LPFC using neuroimaging techniques (such as positron emission tomography [PET] and functional magnetic resonance imaging [fMRI]), lesion studies, and microelectrode recordings in both humans (primarily through neuroimaging and lesion studies) and nonhuman primates. 7 , 8 , 9 , 10 , 11 , 12 , 13 Beginning in the 1930s with the work of Jacobsen, 14 many studies demonstrated through lesion studies in monkeys that the LPFC was essential for delay tasks. Because these tasks (e.g., delayed-response) commonly required memorization of a spatially defined visual stimulus, some researchers initially assumed that the LPFC was the storage site for memory of spatial location. However, later reversible lesion experiment showed that local cooling of LPFC induced temporary deficits in delayed color matching (a nonspatial memory task) that was of the same magnitude as the deficit in delayed response. 15 Furthermore, microelectrode recordings in monkeys performing both tasks revealed spatial and nonspatial memory cells distributed throughout the lateral PFC. 16 Anatomical and physiological evidence has since supported a functional dissociation within the LPFC, dividing it into the dorsolateral PFC (DLPFC) and ventrolateral PFC (VLPFC). 17 , 18 Studies have suggested that the DLPFC receives inputs from the dorsal visual pathway (“where” pathway) originating in the posterior parietal cortex, which is primarily involved in location memory. 19 , 20 , 21 Furthermore, planning tasks consistently activate the DLPFC (PET/fMRI studies), and lesions of this area impair planning. 7 , 8 , 9 , 10 , 11 , 12 , 13 , 22 , 23 , 24 , 25 , 26 , 27 These planning tasks can be considered as elementary operations within the broader working memory system, which maintains and manipulates mental representations to guide future actions by representing stimuli rather than depending on their immediate presence. 20 , 28 Indeed, human activities involving higher cognitive functions such as focused attention, language comprehension and production, reasoning, abstract thinking, planning, and decision-making rely on the integrity of working memory processes. Lesion studies have shown that monkeys with dorsolateral prefrontal damage exhibited chance-level performance on delayed-response tasks, but recovered when the delay is removed, demonstrating that DLPFC is critical for maintaining and manipulating internal representations. 29 , 30 , 31 , 32 Consistently, microelectrode recordings have indicated correlations between DLPFC neural activity and multiple aspects of working memory. 12 , 13 , 33 , 34 , 35 , 36 , 37 , 38 , 39 The VLPFC, in contrast, is suggested to receive projections from the ventral visual pathway (“what” pathway) originating in inferotemporal cortex, thereby specializing in object memory. 19 , 20 , 21 Thanks to its multimodal sensory integration capabilities, 40 the VLPFC combines object characteristics to detect similarities between objects and support abstraction, enabling categorization processes that are crucial across domains of cognition and behavior, from learning to survival. 41 , 42 , 43 Several studies have also highlighted the role of right posterior VLPFC in response inhibition, which can be fractionated into sub processes such as interference control, action withholding, and action cancellation. 44 , 45 Conditional motor learning, or arbitrary visuomotor mapping (e.g., associating a red traffic light with the action of stopping), is essential for rule learning, decision-making, and cognitive control, 46 and although it relies on a large network, the VLPFC is one of the critical nodes in this network. Lesion, electrophysiology, and neuroimaging studies in both monkeys and humans indicate that VLPFC is critical for learning and consolidating associations between stimuli and context-dependent actions, even in the absence of explicit working memory demands. 40 , 47 Furthermore, some other physiological work confirmed this functional segregation, with spatial working memory encoded dorsally and object/face information ventrally. 48 , 49 Importantly, such specialization has been observed even in untrained monkeys, 50 suggesting intrinsic coding properties of PFC neurons. Moreover, Schwartz and Goldman-Rakic 51 showed that much of this connectivity is prenatally specified, indicating that these dorsal-ventral differences are organizational principles rather than products of training. Consistent dissociations have also been demonstrated in human neuroimaging, where spatial versus feature working memory tasks activate distinct prefrontal systems. 52 It must be emphasized, however, that these distinctions are based on the dominant properties of neurons within each region; it may not be entirely accurate to consider the LPFC as two isolated subregions, since both DLPFC and VLPFC participate within a larger distributed network supporting cognitive control and goal-directed behavior. Nevertheless, variability across studies remains, and alternative analytical approaches may help clarify inconsistencies. It has been observed that the dimensionality of spiking activity can be increased or decreased during the deployment of spatial working memory tasks, while the average firing rate of neurons remains unchanged. 53 , 54 Furthermore, it has been proposed that the working memory trace and cognitive functions can be maintained by “dynamic coding” processing using approaches such as computational modeling, i.e., a temporal variability in neuronal activity, particularly in a discontinuous but effective manner. One tool that has not been fully utilized in this field is scale-free patterns, specifically fractals. Given the lack of employing such patterns, their potential applications remain underexplored. Fractal indexes are conceptually very close to biological activities, and many real-world phenomena have some fractal characteristics. 55 In fractal geometry, a fractal dimension (FD) is a statistical index of complexity that looks at the alterations of detail in a pattern (particularly in fractal patterns) as the scale shifts. 56 This scaling can take place in either space or time. For a time series in an n-dimensional space, the FD is a numerical value that can be used to represent how densely the trajectory points fill a subspace as the number of points increases. 57 The FD provides a quantitative measure of time-series complexity. Deviations from integer values (resulting in non-integer FD estimates) reflect denser, more intricate patterns, suggesting enhanced space-filling properties and thus a system with multiscale complexity, with intricate details across multiple scales. 58 , 59 , 60 , 61 These indicators can help to reveal the mechanisms by which brain networks perform different functions, 62 , 63 and can be used as biomarkers. 64 , 65 There are some evidences that fractal patterns can be found in the structure of the CNS and brain. 66 , 67 , 68 Fractal properties can also be seen in various levels of observation, in ion channel dynamics, 69 membrane potential firing rate, 69 , 70 and interaction of neural population. 71 Fractal properties in the interaction between brain regions in cognitive functions are observable, particularly in working memory and decision-making processes. 53 , 72 , 73 , 74 It is worth noting that neural activity in the PFC is known to exhibit high-dimensional, context-dependent dynamics that cannot be fully captured by mean firing rates or conventional decoding analyses. 54 , 75 , 76 While firing rates provide a first-order measure of activity, and decoding approaches assess representational content, they are less sensitive to the temporal irregularity and multi-scale structure of spike trains. FD analysis directly quantifies the degree of scale-free temporal complexity, which has been linked to information integration, cognitive flexibility, and adaptive control. 77 , 78 Moreover, some works have emphasized that scale-free and fractal organization is a fundamental property of neural systems across modalities, providing a principled metric of dynamical richness beyond rate-based summaries. 67 , 68 , 79 In this study, we delve into the fractal properties of prefrontal neuronal firing rates in macaque monkeys using three FD methods of Higuchi (HFD), Katz (KFD), and box-counting (BCFD), exploring their potential as biomarkers for cognitive functions. More specifically, we examined the ability of FDs to detect the content of cognitive functions and working memory from the neural spiking activity of the DLPFC and VLPFC. To this end, we have utilized the publicly available dataset containing the multiunit spiking activity of four monkeys that performed the match-to-sample and working memory task. A cylindrical chamber was implanted on the PFC of the monkeys, and microelectrodes were inserted into the DLPFC and VLPFC for recording while the monkeys performed the task. The recordings were done before training and after training, creating two phases of pre-training and post-training. In the pre-training phase, monkeys were required only to maintain fixation while spatial or feature-based stimuli were presented; no match/non-match rule was yet applied, and animals were not required to perform a memory or decision task. By contrast, the post-training phase consisted of the full match-to-sample paradigm, in which the animals had to observe a sample stimulus, retain it across a delay, and make a match/non-match decision with an eye movement response. Thus, the pre-training condition should be interpreted as passive stimulus-driven activity, while the post-training condition reflects proficient working memory and rule-based decision-making. Figure 1 illustrates the task procedure. Figure 1. Open in a new tab Task procedure and timeline Consecutive frames indicate the experiment setup: (A) the spatial set task in which the monkey is first required to fixate on the location of a white square presented at the center of the monitor. After 1 s, a squared shape stimulus appears in one of the locations in the 3 × 3 grid around the center for 500 ms followed by a 1.5 s delay. Then the second cue (stimulus) is shown in a matched place or diametrical opposite location as non-matched. Following a 1.5 s delay, the choice target appeared in locations orthogonal to the earlier presented stimuli. In this stage, the monkey should specify the match or non-match by saccade toward the blue or green target (only for the post-training phase) to take the reward. (B) In the feature set task, the procedure is the same as the spatial set with the difference that the monkey observes a shape and should specify whether the second cue is the same shape or not, respectively. Here, we have analyzed both the pre-training and post-training phases separately. To compute the FDs, inter-spike intervals (ISIs) were calculated, and the ISIs of individual neurons in different brain regions from four monkeys were pooled across trials for each time interval of the task. The computed FDs should indicate significant meaningful differences between various stages of the task as indicators for the contents of cognitive functions. Specifically, working memory and decision-making following stimulus matching should reflect these differences, since they are the main goal of the designed task. In computing the FDs of ISIs, all the methods provided meaningful and unique interpretations for cognitive functions involved in different task stages, suggesting that they can effectively decode ISI signals and complement one another, although the Katz method could demonstrate greater sensitivity in analyzing the complexity of ISIs among them. The computed FDs revealed significant differences across various time intervals during spatial and feature tasks, highlighting distinct performances of the dorsal and ventral brain regions throughout different training phases. Within the pre-training phase, both regions revealed notable response differences at the first and second delay intervals, particularly in the DLPFC, suggesting sensitivity to changes in stimulus location. In addition, both the dorsal and ventral regions indicated the working memory, stimulus matching, and decision-making contents by varying complexities, thus showing different engagement in the task. Furthermore, comparisons between the tasks and recording regions revealed distinct complexities for both the regions during spatial and feature tasks, specifically in the pre-training phase. These findings could suggest that the complexity difference in the dorsal region between the spatial and feature tasks could reflect the specialization of the spatial processing in the dorsal region. It was observed that the change in the complexity of VLPFC during the second delay interval of the feature task could be attributed to alterations in stimulus shape during this interval. During the post-training phases, few significant changes were observed, suggesting that unique characteristics of the dorsal and ventral regions are more visible before training. Ultimately, the findings highlight the potential of fractal analysis to enhance our understanding of the distinct functions of the DLPFC and VLPFC in processing the spatial and visual characteristics of objects. Additionally, they underscore the influence of training on the functional performance of these brain regions within cognitive contexts. Results To analyze the behavior of the ISI index, we have computed the FD of the pooled ISIs of the monkeys’ trials in different conditions. In fact, the FDs have been calculated based on different types of tasks, recording regions, and three different time intervals (fixation, first delay, and second delay) in the pre-and post-training phases separately. More specifically, for each of the pre-and post-training phases, we considered four different modes within each time period: spatial task in the dorsal region, spatial task in the ventral region, feature task in the dorsal region, and feature task in the ventral region. Note that in the considered dataset, not all task conditions and regions were included for every monkey. Finally, the FD results were compared between the time intervals among different conditions using a two-way analysis of variance (ANOVA) test, considering two factors: time intervals (fixation, delay 1, and delay 2) and training phases in the four modes as mentioned earlier (considering the two training phases, eight conditions were considered as the second way of ANOVA test). For the Higuchi method, a significant interaction effect (F[1,410,587] = 2.81, p = 0.0003) and a significant main effect of factor 2 (F[710,587] = 26.33, p < 0.0001) were observed, whereas the main effect of time interval was not significant (F[210,587] = 0.75, p = 0.47); therefore, within training phase pairwise comparisons for Higuchi should be interpreted cautiously, although they were mostly non-significant. For the Katz method, significant effects were detected for the interaction (F[1,411,505] = 6.21, p < 0.0001), factor 1 (F[211,505] = 92.70, p < 0.0001), and factor 2 (F[711,505] = 40.26, p < 0.0001). Similarly, for the box-counting method, significant interaction (F[146,104] = 3.78, p < 0.0001) and main effects of factor 1 (F[26,104] = 5.18, p = 0.0057) and factor 2 (F[76,104] = 3.19, p = 0.0022) were observed. Although FD metrics primarily reflect changes in the complexity and irregularity of neural signals, variations in complexity observed under specific task conditions and time intervals may reflect changes in cognitive operations associated with the task. In fact, an indicator of the working memory can be identified by observing a significant difference in the calculated complexities between the first delay and its corresponding fixation period during the post-training phase, while there is no significant difference in the pre-training phase. This goal could be achieved by analyzing the FDs of time intervals within the training phases for each recording region of each specific task. Additionally, another indicator of memory content is expected to show a significant difference in the complexity of the ISIs between the pre-training and post-training phases during the first delay interval when no differences should be observed between the conditions during their fixation periods. This could be indicated by analyzing the FDs of time intervals across training phases in each recording region of each specific task. Similarly, to assess rule learning (including stimulus matching and decision-making), a significant complexity difference between the pre-training and post-training phases is expected during the second delay period, while no difference should be observed between the fixation periods. Furthermore, if the second delay period shows no difference from fixation in pre-training but a significant difference in post-training, this could suggest that the monkeys are analyzing the matching of the stimuli and deciding to saccade toward the correct target. To examine the role of the two recording regions in cognitive functions for each task separately, we first presented the mean FDs of ISIs calculated using the Higuchi, Katz, and box-counting methods for the time intervals of both recording regions in the spatial task. Subsequently, the results have been presented for the feature task. Thereafter, we investigated the differences between the two recording regions in each specific task, and finally the differences between the two tasks for each specific region. In addition, we conducted a simulation study using well-known homogeneous Poisson model of stochastic neuronal firing to generate spike trains, 80 in order to show the distinct sensitivities of the three FD methods in ISIs derived from neural data (as being done in this study on real neural data). While this simulation is not part of the primary experiments, it demonstrates how each FD method responds to variations in firing rates and temporal correlations in spike trains. In our simulation, we analyzed the variation of firing rates in spike train and also the temporal correlations using a weighted combination of the previous ISI and a newly drawn exponential ISI with three correlation levels. It should be mentioned that in the context of spike trains, FD of ISIs could quantify the irregularity, scale-invariant variability, and temporal correlations of spike timing, depending on the characteristics and sensitivity of each FD method. Within and between training phases analysis in spatial task Figure 2 presents the mean FDs computed from pooled ISIs of four monkeys, calculated using the Higuchi, Katz, and Box-counting methods (employing the Leibovich-Toth technique) for the spatial task. It displays the FDs for the time intervals during both the pre-training and post-training phases, for DLPFC (first row) and VLPFC (second row) recording areas. To highlight significant changes in FDs among the time intervals within and between the training phases, ANOVA significance levels are marked with the number of asterisk signs in the figures. Multiple comparisons were controlled using Tukey-Kramer; asterisks indicate adjusted p values. Effect sizes are reported by Hedges’ g method. More specifically, in each figure, the asterisks indicate the significant changes in FDs between the time intervals in each training phase and also the FDs of time intervals with their corresponding intervals in the other training phase. Furthermore, Table 1 presents the effect sizes and statistical power for all significant comparisons, indicating medium or close to medium effect sizes and high statistical power across tests. Figure 2. Open in a new tab Fractal dynamics across time intervals in the spatial task Mean FDs of pooled ISIs computed using Higuchi, Katz, and box-counting methods during the spatial task. (A) Dorsolateral prefrontal cortex (DLPFC). (B) Ventrolateral prefrontal cortex (VLPFC). Results are shown for fixation, delay 1, and delay 2 intervals during pre-training and post-training phases. Two-way ANOVA followed by Tukey-Kramer post hoc tests was used to compare time intervals and training phases within each region. Asterisks indicate significant differences (ns = not significant, ∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001, ∗∗∗∗ p < 0.0001); error bars denote 95% confidence intervals. Table 1. Effect size (Hedges’ g) and statistical power for significant FD comparisons presented in Figure 2 across task conditions Metric Contrast Task Area State Epoch Hedges_g Power Significance HFD D2 - Fixation Spatial Dorsal Pre – −0.202850 0.914642 ∗ KFD D2 - Fixation Spatial Dorsal Pre – 0.335795 0.999938 ∗∗∗∗ KFD D1 - Fixation Spatial Dorsal Pre – 0.322012 0.999851 ∗∗∗ KFD Post - Pre Spatial Dorsal – Delay 2 −0.211100 0.949092 ∗ KFD D2 - Fixation Spatial Ventral Post – 0.310735 0.981832 ∗∗ KFD D2 - Fixation Spatial Ventral Pre – 0.508439 1 ∗∗∗∗ KFD D1 - Fixation Spatial Ventral Pre – 0.547376 1 ∗∗∗∗ KFD Post - Pre Spatial Ventral – Delay 1 −0.215370 0.837917 ∗ BCFD D2 - Fixation Spatial Dorsal Post – 0.271414 0.918019 ∗ BCFD Post - Pre Spatial Dorsal – Fixation −0.450650 0.999483 ∗∗∗∗ Open in a new tab Within training phases In Figure 2 , the FDs obtained using the Higuchi method ranged from 1.904 to 2.039, showing no significant changes across time intervals within either the pre-training or post-training phases for both the DLPFC and VLPFC. In contrast, the Katz method revealed FD values ranging from 1.000 to 1.014, with several significant changes across time intervals in both the DLPFC and VLPFC. Notably, significant increases in FDs were observed during the first and second delay intervals compared to their corresponding fixation periods in the pre-training phase of both recording regions ( p < 0.001). These complexity variations could suggest that both the DLPFC and VLPFC play a role as the monkeys engaged in the spatial task during the delays, indicating heightened cognitive engagement and possibly more complex neural representations of task-related information during the pre-training phase. However, this “heightened engagement” should not be interpreted as evidence of cognitive control or rule-based working memory, since the animals were not yet performing an active decision task. During pre-training, monkeys only maintained fixation while stimuli were passively presented; thus, the observed FD changes likely reflect stimulus-driven neural responsiveness (for example, attentional orienting or sensory adaptation to changes in stimulus location) rather than explicit working-memory encoding. This interpretation aligns with previous studies showing that prefrontal neurons, including those in both DLPFC and VLPFC, can exhibit stimulus-selective and delay-period activity even under passive viewing conditions. 20 , 21 , 48 , 49 , 50 In this context, “heightened cognitive engagement and complex neural representations” refer specifically to the stimulus-driven responses rather than to rule learning or task performance. Throughout the rest of the article, the terms “increased cognitive engagement” or “complex neural representation” in the pre-training phase, refer to stimulus-driven modulation of activity within dorsal-ventral networks rather than explicit cognitive control. In the post-training phase, the Katz method showed an increase in FDs during the second delay interval for VLPFC recordings compared to the corresponding fixation period ( p < 0.001). This change could suggest functional differences in the VLPFC region for the second delay time interval; however, since the same variation was observed in the pre-training phase, it remains inconclusive whether the content of decision-making is affected. The box-counting method on the other hand exhibited a wider range of the FD values (from 0.619 to 2.336) and could only reveal one significant change during the within-training phase investigation. Specifically, in the post-training phase, the FD increased in the DLPFC between the fixation and second delay intervals ( p < 0.05). This finding suggests that the engagement of the DLPFC varies as the task progresses, potentially influenced by its role in processing visual stimuli, particularly given that the spatial locations of successive stimuli could vary. Note that match and non-match trials were pooled for the present analyses to ensure sufficiently long and statistically stable ISI sequences for FD estimation. FD methods such as Higuchi and box-counting require long, continuous time series to yield reliable estimates, which could not be achieved by subdividing trials further. This pooling approach allows the average effect of spatial variation (which occurs in non-match trials with maximal displacement) to be reflected in the mean FD values, although it likely attenuates the magnitude of this difference. Future studies with datasets containing a larger number of trials per condition and explicit match/non-match labeling should examine whether FD changes scale systematically with spatial displacement or choice outcome, to more precisely dissociate stimulus-driven from rule-based effects. Results from within-training phase analyses of the spatial task highlight the dynamic engagement of the DLPFC and VLPFC during cognitive tasks. Firstly, KFD revealed that both DLPFC and VLPFC showed significant differences across epochs (delay 1 and delay 2 vs. fixation) during the pre-training phase, whereas in the post-training phase, only VLPFC exhibited a significant delay2-fixation difference; however, the presence of similar variations in the pre-training phase raises questions about the incorporation of decision-making content in the VLPFC. These results indicate that time epoched distinctions were more obvious in pre-training (it should be noted that the functional comparisons between dorsal and ventral regions across training phases are presented separately in Figure 4 ). Finally, the box-counting method reveals a significant change in the FDs of the DLPFC between the fixation and second delay intervals of the post-training phase, suggesting the responsiveness of DLPFC to changes in stimulus location. This observation aligns with prior findings from the same dataset showing that the number of selective neurons in both regions decreased after training, with a higher ratio of selective neurons remaining in the dorsal region compared to the ventral region during the spatial task. 81 Figure 4. Open in a new tab Dorsal-ventral differences during the spatial task The mean FDs of the ISI signals calculated using the Higuchi, Katz, and box-counting methods across different time intervals during both training phases of the spatial task. Subfigures show the results for the dorsal and ventral regions side by side for comparison. The first row illustrates the FDs obtained from the Higuchi method, the second row presents results from the Katz method, and the final row demonstrates the outcomes from the box-counting method. Furthermore, the results of the two-way ANOVA followed by Tukey-Kramer post hoc were presented by asterisk signs to indicate the significant differences between time intervals within identical training phases across different recording regions (ns = not significant, ∗∗ p < 0.01, ∗∗∗ p < 0.001). The error bars indicate 95% CIs. It is also worth mentioning that the key distinction between the pre-training and post-training phases lies in their behavioral demands. During pre-training, the monkeys were required only to maintain fixation while passively viewing stimuli, whereas during post-training, they used the same spatial or object information to guide match versus non-match decisions. This difference in task context fundamentally changes the functional meaning of neural activity. Prior studies have demonstrated this principle clearly: Ó Scalaidhe et al. 50 showed that neurons in the VLPFC exhibit stimulus selectivity even under passive viewing, yet their activity becomes strongly modulated when the same stimuli are used in an active working-memory task. These findings emphasize that the neural representation of identical stimuli depends critically on task demands and behavioral goals. Accordingly, although spatial variations were present in both training phases, their functional implications might differ. In pre-training, FD differences likely reflect stimulus-driven or attention-related responses, whereas in post-training they are shaped by rule-based working-memory and decision-making processes. It should also be mentioned that FD metrics primarily reflect changes in the complexity and irregularity of neural signals rather than specific cognitive operations. Accordingly, significant FD modulation in certain conditions is interpreted as evidence for changes in neural complexity, whereas the absence of FD differences in other conditions does not imply a lack of cognitive or functional involvement. Between training phases When comparing the time intervals between the pre-training and post-training phases, the Higuchi method indicated no significant changes in FDs of both dorsal and ventral region between the pre- and post-training phases. Notably, VLPFC in Higuchi exhibits a downward shift from pre- to post-trainning phases across all three epochs; although, not significant. The Katz method on the other hand displays a significant change (decrease) in the FD of the second delay interval during the post-training phase compared to the pre-training phase ( p < 0.0001) for the DLPFC region. In addition, it showed altered ISI complexity for VLPFC region during the first delay in post-training. Notably, these changes occurred without significant differences between fixation periods across training phases. In contrast, the box-counting method displays a significant change (decrease) in the FD of the fixation interval during the post-training phase compared to the pre-training phase ( p < 0.0001) for the DLPFC region. Here, we can observe that the trend of the first and second delay related to the fixation changed from pre-training to post-training and thus it can suggest a difference in the neural response of the DLPFC region before and after the training. Consequently, the changes observed in complexity during the first delay time interval between the two training phases could suggest the presence of working memory contents during the spatial task in both regions. In addition, the complexity changes during the second delay interval between the two training phases in the dorsal and the ventral regions could suggest the presence of stimulus matching and decision-making content in both regions during the spatial task. Since the monkeys exhibited proficient task performance during the post-training phase, these suggestions regarding working memory and decision-making or rule-learning processes could be expressed. Please note that these findings are method-dependent and do not converge across all three methods. Therefore, they should be considered cautiously as suggestive evidence that working memory demands emerge in the post-training first delay. Within and between training phases analysis in feature task Figure 3 demonstrates the mean FDs calculated from the pooled ISIs of four monkeys, using the Higuchi, Katz, and box-counting methods for the feature task. Similar to Figure 2 , it presents the FDs across different time intervals during both the pre-training and post-training phases for both the DLPFC (first row) and VLPFC (second row) recording areas. To emphasize significant changes in FDs among the time intervals within and between the training phases, the significance levels from the two-way ANOVA test are represented by the number of asterisks in the figures. Multiple comparisons were controlled using Tukey-Kramer; asterisks indicate adjusted p values. Effect sizes are reported by Hedges’ g method. Specifically, the asterisks indicate significant changes in FDs between the time intervals in each training phase, as well as between the time intervals and their equivalents in the other training phase. Figure 3. Open in a new tab Fractal dynamics across time intervals in the feature task The mean FDs of the ISIs calculated using Higuchi, Katz, and box-counting methods for the feature task. (A) Represents the FDs of the dorsal region during the feature task for pre-training (hatched periwinkle) and post-training (hatched charcoal) phases. (B) Represents the FDs of the ventral region during the feature task for pre-training (hatched light pink) and post-training (hatched burgundy) phases. The horizontal axes of subfigures represent three different modes of time intervals: fixation, delay 1, and delay 2 for two training phases. The vertical axis represents the FD calculated using the three methods. The results of the two-way ANOVA followed by Tukey-Kramer post hoc were presented to compare time intervals and training phases for each region separately. The significant differences between the time intervals in each specific condition of the feature task are indicated by asterisks (ns = not significant, ∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗∗ p < 0.0001). The error bars indicate 95% CIs. Furthermore, Table 2 presents the effect sizes and statistical power for all significant comparisons, indicating medium or close to medium effect sizes and high statistical power across tests. Table 2. Effect size (Hedges’ g) and statistical power for significant FD comparisons presented in Figure 3 across task conditions Metric Contrast Task Area State Epoch Hedges_g Power Significance KFD D2 - Fixation Feature Dorsal Pre – 0.506867 1 ∗∗∗∗ KFD D1 - Fixation Feature Dorsal Pre – 0.468931 1 ∗∗∗∗ KFD Post - Pre Feature Dorsal – Delay 1 −0.268950 0.995697 ∗∗ KFD Post - Pre Feature Dorsal – Delay 2 −0.358070 0.999984 ∗∗∗∗ KFD D2 - Fixation Feature Ventral Post – 0.265900 0.915655 ∗ KFD D2 - Fixation Feature Ventral Pre – 0.300662 0.997275 ∗∗∗∗ KFD D1 - Fixation Feature Ventral Pre – 0.408757 0.999996 ∗∗∗∗ KFD Post - Pre Feature Ventral – Delay 1 −0.320750 0.994282 ∗∗∗∗ Open in a new tab Within training phases For the feature task, Figure 3 indicates that within the training phases, neither the Higuchi nor box-counting methods showed significant FD differences across time intervals in either the DLPFC or VLPFC region. In contrast, the Katz method identified several significant changes in FDs across time intervals during the pre-training phase in both the DLPFC and VLPFC regions. Specifically, the Katz method revealed significant differences in FDs of the first and second delay intervals compared to their fixation in the pre-training phase in both regions. Similar to spatial tasks, these significant differences could indicate the difference in cognitive engagement for both regions in the pre-training phase. Moreover, the Katz method also detected a significant difference between the second delay and fixation epochs in the post-training phase of VLPFC. The pronounced difference in significance levels for this contrast between the pre- and post-training phases suggests that the underlying cognitive processes differ across stages, reflecting a transition from passive, stimulus-driven engagement in the pre-training phase to rule-based decision-related activity after training. As a result, in the within-training phase investigation of the feature task, only the Katz method revealed significant changes for the first and second delay intervals during the pre-training phase in both regions and also for the second delay interval during post-training phase of VLPFC. During pre-training phase, since FD changes were observed in both spatial and feature tasks, but in response to different types of stimulus variation (location vs. object characteristics), these results may reflect distinct task-related processes. In addition, since no significant differences were mostly found between the time intervals during the post-training phase, we can suggest from the FD measure point of view that the specific characteristics of the dorsal and ventral are more distinguishable before training. These findings align with the observed decrease in selective neurons in both the dorsal and ventral regions following the learning process. 81 Please note that these interpretations are cautiously mentioned and that our analysis can only demonstrate the presence of FD changes, not their mechanistic origins although the results showed consistency with the prior literature. Between training phases When comparing the pre-training and post-training phases, significant alterations in complexity were observed by the Katz method during the first and second delay periods of the dorsal and ventral (only first delay epoch) regions. Similar to the results of spatial task, since there is no significant change between the fixation periods of the two training phases, it can be suggested that the variations in the delay intervals are related to changes in the cognitive and neural contents. Consequently, the changes observed in complexity between the two training phases during the first delay time interval and second delay time intervals in the dorsal region suggest the presence of working memory, stimulus matching, and decision-making contents in the DLPFC recording areas while the monkeys performed the feature task. In addition, the variations in the first delay interval of VLPFC could be related to the content of working memory too. Overall, by observing the results from the FD measure point of view, presented in within and between training phases analysis of the spatial and feature tasks, we can suggest that both the DLPFC and VLPFC are engaged during the working memory and decision-making stages of the tasks, consistent with the literature. Based on previous literature, it can be suggested that these regions contribute differently to these processes: the DLPFC is more closely associated with the maintenance and manipulation of working memory contents, whereas the VLPFC is more involved in decision-making and rule-based processing. This explanation aligns with established evidence indicating a functional dissociation within the LPFC, where dorsolateral regions primarily support the retention of task-relevant information and ventrolateral regions facilitate stimulus evaluation and rule application. Dorsal versus ventral region in spatial task Figure 4 presents the FDs for each time interval in the dorsal and ventral recording regions to examine the sensitivity of the three FD methods in detecting complexity changes between the recording regions during the spatial task. In addition, the two-way ANOVA interaction results have been presented (marked with asterisks) to indicate FD variations between different recording regions under matching time intervals and training conditions. Multiple comparisons were controlled using Tukey-Kramer; asterisks indicate adjusted p values. Effect sizes are reported by Hedges’ g method. To avoid redundancy, we excluded significant changes within and between the pre-and post-training phases of each region, as these results were previously presented in Figure 2 . Instead, we have focused on the significant differences between time intervals within the matching training phase across the different recording regions. Furthermore, Table 3 presents the effect sizes and statistical power for all significant comparisons presented in Figure 4 , indicating medium or close to medium effect sizes and high statistical power across tests. Table 3. Effect size (Hedges’ g) and statistical power for significant FD comparisons presented in Figure 4 across task conditions Metric Contrast Task Area State Epoch Hedges_g Power Significance HFD Ventral - Dorsal Spatial – Pre Delay 2 0.287500 0.990357 ∗∗ HFD Ventral - Dorsal Spatial – Pre Delay 1 0.324295 0.998096 ∗∗∗ KFD Ventral - Dorsal Spatial – Post Delay 2 0.319046 0.996620 ∗∗∗ KFD Ventral - Dorsal Spatial – Pre Delay 2 0.251042 0.975224 ∗∗ KFD Ventral - Dorsal Spatial – Pre Delay 1 0.298513 0.996408 ∗∗∗ Open in a new tab Figure 4 indicates that the Higuchi and Katz methods show differences between the pre-training phases of the dorsal and ventral regions during the first and second delay intervals. In addition, Katz method shows a difference between the second delay intervals of the dorsal and ventral regions in post-training phase. It is worth noting that previous results indicated both regions are involved during the time intervals of each training phase and across training phases. Therefore, this significant difference between the two regions could suggest that the dorsal and ventral regions are processing different contents with varying complexities during the spatial task. For example, the dorsal region might be more engaged in understanding the location of the stimulus (during the first delay interval) and in comparing the location of the second stimulus with the first (during the second delay interval). In contrast, the ventral region may focus more on the details of the stimulus shape, striving to adhere to the task rules (in the interval after the first stimulus display) and then deciding on the appropriate action based on learned rules (such as saccade toward the correct target) according to whether the stimuli are matched or not (during the second delay interval). Dorsal versus ventral region in feature task Figure 5 presents the FDs within each time interval across different recording regions to examine the sensitivity of the three FD methods in detecting complexity changes between the recording regions during the feature task. In addition, the multiple comparison results of two-way ANOVA using Tukey-Kramer post hoc have been presented (marked with asterisks) to indicate FD variations between different recording regions under matching time intervals and training conditions. Figure 5. Open in a new tab Dorsal-ventral differences during the feature task The mean FDs of the ISI signals calculated using the Higuchi, Katz, and box-counting methods across different time intervals during both training phases of the feature task. Subfigures show the results for the dorsal and ventral regions side by side for comparison. The first row illustrates the FDs obtained from the Higuchi method, the second row presents results from the Katz method, and the final row demonstrates the outcomes from the box-counting method. Furthermore, the results of the two-way ANOVA followed by Tukey-Kramer post hoc were presented by asterisks to indicate the significant differences between time intervals within the same training phases across different recording regions (ns = not significant, ∗ p < 0.05, ∗∗∗ p < 0.001, ∗∗∗∗ p < 0.0001). The error bars indicate 95% CIs. To avoid redundancy, we excluded significant changes within and between the pre- and post-training phases of each region, as these results were previously shown in Figure 3 . Instead, we have focused exclusively on the significant differences between time intervals within the matching training phase across the different recording regions. Furthermore, Table 4 presents the effect sizes and statistical power for all significant comparisons presented in Figure 5 , indicating medium or close to medium effect sizes and high statistical power across tests. Table 4. Effect size (Hedges’ g) and statistical power for significant FD comparisons presented in Figure 5 across task conditions Metric Contrast Task Area State Epoch Hedges_g Power Significance HFD Ventral - Dorsal Feature – Post Delay 2 0.433165 0.999970 ∗∗∗∗ HFD Ventral - Dorsal Feature – Pre Delay 2 0.367803 0.999959 ∗∗∗∗ HFD Ventral - Dorsal Feature – Pre Delay 1 0.454214 1 ∗∗∗∗ HFD Ventral - Dorsal Feature – Pre Fixation 0.323496 0.998523 ∗∗∗ KFD Ventral - Dorsal Feature – Post Delay 2 0.508690 1 ∗∗∗∗ KFD Ventral - Dorsal Feature – Post Delay 1 0.278832 0.977260 ∗ KFD Ventral - Dorsal Feature – Pre Delay 2 0.212049 0.943275 ∗ KFD Ventral - Dorsal Feature – Pre Delay 1 0.353984 0.999961 ∗∗∗∗ KFD Ventral - Dorsal Feature – Pre Fixation 0.418204 0.999999 ∗∗∗∗ BCFD Ventral - Dorsal Feature – Post Delay 2 0.350002 0.959536 ∗ Open in a new tab Figure 5 shows that, similar to the spatial task during the pre-training phase, both the Higuchi and Katz methods detected differences in complexity between the dorsal and ventral regions in the feature task. However, here the variations in complexity revealed by these methods exhibited different significance levels between the fixation periods and the two delay intervals. Therefore, we cannot conclusively attribute the observed variability in FD differences during the delay intervals to differential engagement of these regions in working memory, decision-making, or rule-learning processes, although such functions may still contribute to the observed patterns. In the post-training phase, all three FD methods revealed significant differences between the dorsal and ventral regions during the second delay interval. In addition, the Katz method showed a significant change in complexity between the two regions during the first delay of the post-training phase. Similar to the spatial task, previous results suggest that both regions are active across time intervals within and between training phases. Thus, the significant differences observed between dorsal and ventral regions likely reflect their involvement in processing distinct types of information (with varying degrees of complexity) during the feature task. Spatial versus feature task in dorsal region To survey the complexity changes for each time interval across different task sets within each specific area of recording, the computed FDs for each time interval of the two tasks in the dorsal region and also their corresponding interaction results of the ANOVA test have been presented in Figure 6 . Figure 6. Open in a new tab Spatial versus feature task comparison in the dorsal region The mean FDs of the ISI signals, calculated using the Higuchi, Katz, and box-counting methods, are presented across different time intervals during the training phases of both tasks within the dorsal region. Each subfigure displays the results for the spatial and feature tasks side by side to facilitate comparison between the tasks in the dorsal region. The first row illustrates the FDs obtained from the Higuchi method, the second row presents results from the Katz method, and the final row demonstrates the outcomes from the box-counting method. Furthermore, asterisks indicate significant results of the two-way ANOVA, followed by Tukey-Kramer post hoc, showing differences between time intervals within identical training phases across the tasks (ns = not significant, ∗ p < 0.05). The Hedges’ g effect size and the statistical powers are presented along with the asterisk sign. The error bars indicate the 95% CIs. Regarding Figure 6 , among the three FD estimation methods, only the box-counting approach revealed a significant difference ( p < 0.05) between the two task types in the dorsal region during the fixation period. Here, the changes in FD during the two delay intervals relative to fixation did not reach statistical significance; however, their trends and the statistically significant change in the fixation period (particularly during the pre-training phase, when stimulus locations varied and the monkeys had not yet learned the task rules) might reflect the greater spatial sensitivity of the dorsal region noted in the previous literature. Spatial versus feature task in ventral region To survey the complexity changes for each time interval across different task sets within the ventral region, the corresponding FDs and their interaction results of the ANOVA test are presented in Figure 7 . Figure 7. Open in a new tab Spatial versus feature task comparison in the ventral region The mean FDs of the ISI signals, calculated using the Higuchi, Katz, and box-counting methods, are presented across different time intervals during the training phases of both tasks within the ventral region. Each subfigure displays the results for the spatial and feature tasks side by side to facilitate comparison between the tasks in the ventral region. The first row illustrates the FDs obtained from the Higuchi method, the second row presents results from the Katz method, and the final row demonstrates the outcomes from the box-counting method. Furthermore, the results of the two-way ANOVA followed by Tukey-Kramer post hoc are presented by asterisk signs to indicate the significant differences between time intervals within identical training phases across different tasks (ns = not significant, ∗ p < 0.05). The Hedges’ g effect size and the statistical powers are presented along with the asterisk signs. The error bars indicate the 95% CIs. Similarly, the results from the three FD calculation methods shown in Figure 7 indicate statistically significant changes in complexity for different time intervals during the pre-training phase for the ventral region. These changes are consistent with the previous literature in the differences in the role of the ventral region between spatial tasks and feature tasks, particularly for the second delay interval. Specifically, the Higuchi method detected the change in complexity for the delay interval after the presentation of the second stimulus (delay 2), while the Katz method could recognize the difference for the fixation period. The complexity differences between the spatial and feature tasks in the ventral region during the second delay interval of the pre-training phase could indicate varying performance in this region for both tasks. Given that the ventral region is thought to be involved in analyzing object characteristics, this change in complexity can be attributed to the alterations in stimulus shape during the second delay interval of the feature task. The results and trends observed in Figures 4 and 7 suggest that FD measures capture differences in neural signal complexity across experimental conditions, phases, and recording regions. More specifically, the FD results could strengthen the evidence for differential regional roles, especially when used alongside other neural decoding methods, and could more clearly show that the dorsal region emphasizes the locations where the stimuli are presented, while the ventral region emphasizes the features and visual similarities of the stimuli. After training, although both regions were active during the tasks, there were no significant differences in the complexity of the signals recorded from these areas. This is consistent with the observed decrease in selective neurons in both the dorsal and ventral regions following the learning process. 81 It is worth noting that both DLPFC and VLPFC regions showed sensitivity to the task types (feature and spatial), but through our investigation and comparison of the two regions in one specific task and the two tasks in one specific region, it was suggested that the DLPFC exhibited greater sensitivity to spatial changes, whereas the VLPFC responded more to feature or object variations (patterns that align with established anatomical projections from dorsal and ventral visual streams). Simulation study Poisson spike trains simulation with different firing rates In our simulation, spike trains were generated with firing rates systematically varied between 4 and 7 Hz to investigate the effect of firing rate on the FDs for 1.5 s time length, the same as the time length in delay intervals in the considered task. Figure 8 illustrates how dependent are the estimated FDs computed with three methods of Higuchi, Katz, and Box-counting to the firing rate in simulated spike trains. For each condition, spike trains were generated over 30 repetitions (the minimum required number of observations to support Gaussian distribution while keeping the effect size of statistical test high enough), and FD values were computed. The bars represent the mean FD values across repetitions, while the error bars denote the 95% confidence intervals (CIs), reflecting variability in the estimator. For each FD method separately, we tested whether FD differed across firing rates (four levels: 4–7 Hz) with a one-way ANOVA. In the one-way ANOVA, significant effects of firing rate were observed for the Katz method (F[3,116] = 3,353.50, p < 1 × 10 −8 ) and the box-counting method (F[3,116] = 3.01, p = 0.033), whereas the effect did not reach statistical significance for the Higuchi method (F[3,116] = 2.39, p = 0.073). Post hoc pairwise comparisons were performed using the Tukey-Kramer correction for multiple comparisons. Table 5 presents the effect sizes and statistical power for all significant comparisons presented in Figure 8 , indicating high effect sizes and statistical powers across tests. Figure 8. Open in a new tab Effect of firing rate on fractal dimension estimates in simulated spike trains FD values of ISI signals across different firing rates, averaged over 30 repetitions, using three methods: (A) Higuchi, (B) Katz, and (C) box-counting. Error bars represent 95% CIs. Results of multiple comparisons from a one-way ANOVA with Tukey-Kramer post hoc test are shown above each pair, indicating the levels of statistical significance (ns = not significant, and ∗∗∗∗ p < 0.0001). Higuchi (A) detected no significant difference in firing rate changes, Katz (B) revealed a consistent inverse relation with firing rate due to its sensitivity to signal amplitude, and box-counting (C) did not show significant differences across firing rates of ISI signals. Table 5. Effect size (Hedges’ g) and statistical power for significant FD comparisons presented in Figure 8 across different firing rates of simulation study Metric Contrast (firing rate [Hz]) Hedges_g Power Significance KFD 4–5 8.661666 1 ∗∗∗∗ KFD 4–6 16.359785 1 ∗∗∗∗ KFD 4–7 21.397593 1 ∗∗∗∗ KFD 5–6 8.813452 1 ∗∗∗∗ KFD 5–7 15.498834 1 ∗∗∗∗ KFD 6–7 9.046928 1 ∗∗∗∗ Open in a new tab The x axis within each subfigure, represent the firing rates (4–7 Hz) related to different FD methods. This visualization allows direct comparison of how estimated FDs vary across firing rates. The FD analysis of the ISI signals across different firing rates demonstrated method-dependent sensitivities. The Higuchi method was unable to distinguish firing rate differences. This pattern is consistent with Higuchi’s theoretical properties, as it is primarily responsive to temporal correlations and long-range dependencies; thus, rate-driven differences alone produce minimal changes in memory less Poisson ISIs. In contrast, the Katz method revealed firing rate differences consistently and in a reverse relation: higher firing rates produced smaller ISIs (lower amplitude in the ISI signal), resulting in lower FD values, whereas lower firing rates with larger ISIs yielded higher FD values. This reflects Katz’s strong dependence on signal amplitude scaling. The box-counting method, however, was unable to detect meaningful differences across firing rates, consistent with its lower sensitivity to amplitude variations and its reliance on global geometric measure of space-filling complexity rather than local temporal fluctuations. As suggested in prior works, Box-counting and Higuchi are expected to be more responsive to temporal correlations and scaling irregularities in the signal, while Katz is more directly affected by amplitude. This reasoning motivated our additional simulations using temporally correlated ISI signals to further evaluate the differential sensitivities of these methods. These findings confirm that clear high sensitivity of Katz method in our empirical data could reflect its specific responsiveness to irregularity caused by firing rate changes. Poisson spike trains simulation with serially correlated ISIs Since real neuronal spike trains often deviate from ideal Poisson statistics by exhibiting history dependence and temporal correlations, we extended our simulations to incorporate serially correlated ISIs. The results are summarized in Figure 9 where the first column presents statistical comparisons across firing rates within each correlation level (ρ = 0, 0.3, 0.6), and the second column presents comparisons across correlation levels for each firing rate. A two-way ANOVA was conducted separately for each FD estimator, with firing rate and correlation strength as fixed factors. For the Higuchi method, significant main effects of firing rate (F[3,348] = 293.3, p < 0.0001) and correlation strength (F[2,348] = 1,965, p < 0.0001), as well as a significant interaction effect (F[6,348] = 95.12, p < 0.0001), were observed. Similarly, for the Katz method, significant main effects of firing rate (F[3,348] = 18,246, p < 0.0001) and correlation strength (F[2,348] = 22,039, p < 0.0001), together with a significant interaction (F[6,348] = 138.2, p < 0.0001), were detected. For the box-counting method, the main effect of firing rate was significant (F[3,348] = 5.009, p = 0.0021), as was the main effect of correlation strength (F[2,348] = 4,460, p < 0.0001), whereas the interaction effect was not significant (F[6,348] = 1.105, p = 0.3590). Post hoc pairwise comparisons were performed using the Tukey-Kramer correction for multiple comparisons. Tables 6 and 7 present the corresponding effect sizes (Hedges’ g) and estimated statistical power for the pairwise comparisons. Table 6 shows comparisons across firing rates within each correlation level, while Table 7 shows comparisons across correlation levels within each firing rate. Overall, the results indicate high effect sizes and strong statistical power across tests. The comparisons were reported in two complementary forms: (i) firing-rate differences within each correlation level and (ii) correlation-level differences within each firing rate. Figure 9. Open in a new tab Effects of firing rate and temporal correlation on fractal dimensions FD results for ISI sequences simulated at different firing rates (4–7 Hz) and temporal correlations (ρ = 0, 0.3, 0.6). Rows correspond to (A) Higuchi, (B) Katz, and (C) Box-counting methods. Bars indicate mean FD ± 95% CI across 30 repetitions. The two-way ANOVA significance levels (corrected with Tukey-Kramer correction) are indicated above each pair (ns = not significant, ∗ p < 0.05, ∗∗∗∗ p < 0.0001). The left column shows post hoc comparisons across firing rates within each correlation group, while the right column shows comparisons across correlation groups for each firing rate. Results confirm that KFD is primarily sensitive to firing rate (inverse relation), HFD reflects rate differences only when temporal correlations are present, and BCFD is most sensitive to correlation strength. Table 6. Effect size (Hedges’ g) and statistical power for significant FD comparisons presented in Figure 9 across different firing rates within each correlation level of simulation study Metric Correlation level Contrast (firing rate [Hz]) Hedges_g Power Significance HFD 3 4–5 2.328453 1 ∗∗∗∗ HFD 3 4–6 2.223390 1 ∗∗∗∗ HFD 3 4–7 4.070754 1 ∗∗∗∗ HFD 3 5–7 1.881894 1 ∗∗∗∗ HFD 3 6–7 2.775152 1 ∗∗∗∗ HFD 6 4–5 3.224793 1 ∗∗∗∗ HFD 6 4–6 5.261700 1 ∗∗∗∗ HFD 6 4–7 8.943533 1 ∗∗∗∗ HFD 6 5–6 2.599340 1 ∗∗∗∗ HFD 6 5–7 5.925877 1 ∗∗∗∗ HFD 6 6–7 2.083934 0.999946 ∗∗∗∗ KFD 0 4–5 8.661666 1 ∗∗∗∗ KFD 0 4–6 16.359785 1 ∗∗∗∗ KFD 0 4–7 21.397593 1 ∗∗∗∗ KFD 0 5–6 8.813452 1 ∗∗∗∗ KFD 0 5–7 15.498834 1 ∗∗∗∗ KFD 0 6–7 9.046928 1 ∗∗∗∗ KFD 3 4–5 14.794398 1 ∗∗∗∗ KFD 3 4–6 32.645428 1 ∗∗∗∗ KFD 3 4–7 40.392861 1 ∗∗∗∗ KFD 3 5–6 19.736577 1 ∗∗∗∗ KFD 3 5–7 29.846980 1 ∗∗∗∗ KFD 3 6–7 18.608928 1 ∗∗∗∗ KFD 6 4–5 16.862165 1 ∗∗∗∗ KFD 6 4–6 29.270709 1 ∗∗∗∗ KFD 6 4–7 38.196773 1 ∗∗∗∗ KFD 6 5–6 20.111495 1 ∗∗∗∗ KFD 6 5–7 37.083669 1 ∗∗∗∗ KFD 6 6–7 15.772040 1 ∗∗∗∗ Open in a new tab Table 7. Effect size (Hedges’ g) and statistical power for significant FD comparisons presented in Figure 9 across different correlation levels within each firing rate of simulation study Metric Firing rate (Hz) Contrast (correlation level) Hedges_g Power Significance HFD 4 0–3 −3.768854 1 ∗∗∗∗ HFD 4 0–6 0.724113 0.581631 ∗ HFD 4 3–6 4.909965 1 ∗∗∗∗ HFD 5 0–3 −2.486913 1 ∗∗∗∗ HFD 5 0–6 3.657424 1 ∗∗∗∗ HFD 5 3–6 7.302497 1 ∗∗∗∗ HFD 6 0–3 −3.463294 1 ∗∗∗∗ HFD 6 0–6 5.656187 1 ∗∗∗∗ HFD 6 3–6 9.892709 1 ∗∗∗∗ HFD 7 0–3 −1.884066 0.999911 ∗∗∗∗ HFD 7 0–6 9.682578 1 ∗∗∗∗ HFD 7 3–6 14.287286 1 ∗∗∗∗ KFD 4 0–3 4.705741 1 ∗∗∗∗ KFD 4 0–6 18.418058 1 ∗∗∗∗ KFD 4 3–6 20.584704 1 ∗∗∗∗ KFD 5 0–3 6.318171 1 ∗∗∗∗ KFD 5 0–6 24.716643 1 ∗∗∗∗ KFD 5 3–6 27.697048 1 ∗∗∗∗ KFD 6 0–3 14.748953 1 ∗∗∗∗ KFD 6 0–6 36.015614 1 ∗∗∗∗ KFD 6 3–6 38.804933 1 ∗∗∗∗ KFD 7 0–3 13.113506 1 ∗∗∗∗ KFD 7 0–6 35.221817 1 ∗∗∗∗ KFD 7 3–6 49.040793 1 ∗∗∗∗ BCFD 4 0–3 3.627158 1 ∗∗∗∗ BCFD 4 0–6 11.501999 1 ∗∗∗∗ BCFD 4 3–6 19.545223 1 ∗∗∗∗ BCFD 5 0–3 6.311081 1 ∗∗∗∗ BCFD 5 0–6 22.541728 1 ∗∗∗∗ BCFD 5 3–6 39.149215 1 ∗∗∗∗ BCFD 6 0–3 2.672828 1 ∗∗∗∗ BCFD 6 0–6 29.191047 1 ∗∗∗∗ BCFD 6 3–6 5.330557 1 ∗∗∗∗ BCFD 7 0–3 2.911196 1 ∗∗∗∗ BCFD 7 0–6 32.300983 1 ∗∗∗∗ BCFD 7 3–6 5.351592 1 ∗∗∗∗ Open in a new tab In the within-correlation analyses (left column), the results showed that HFD was insensitive to firing rate under pure Poisson conditions (ρ = 0), but once temporal correlations were introduced, it became increasingly sensitive to firing rate differences, showing an inverse dependence between FD and rate. This confirms that Higuchi primarily detects temporal dependencies and only reflects amplitude-driven differences when correlations are presented. KFD, by contrast, consistently tracked firing rate across all correlation levels, maintaining its known inverse relationship with firing rate due to its sensitivity to ISI amplitude and local irregularity. BCFD remained insensitive to firing rate regardless of correlation level, indicating that this estimator does not capture amplitude differences in 1D ISI signals. In the between-correlation analyses (right column), all three methods detected significant effects of temporal correlation, however, with different magnitudes. BCFD showed the strongest sensitivity, with marked decreases in FD as correlation increased, reflecting its global sensitivity to reduced geometric spread when ISIs become more predictable. HFD also decreased systematically with correlation, consistent with its design to capture temporal scaling and serial dependencies. KFD was also affected, though still showed significant modulation to firing rate (amplitude) changes, indicating that local irregularity is also reduced by correlations. Discussion Our analyses show that FD can reveal distinct patterns of complexity modulation and differentiate neural dynamics in the DLPFC and VLPFC. In pre-training, FD changes were consistently observed in both dorsal and ventral regions even without explicit rule learning. These results align with previous findings that prefrontal neurons exhibit selectivity for spatial and object information even in untrained monkeys or during passive viewing conditions. 48 , 49 , 50 Importantly, Goldman-Rakic demonstrated that the dorsal visual pathway projects to DLPFC and supports spatial representations, while the ventral pathway projects to VLPFC and supports feature/object representations. 20 , 21 Moreover, Schwartz et al., 51 showed that much of this connectivity is prenatally specified, explaining why dorsal-ventral specializations are detectable even prior to training. Thus, our pre-training FD results likely reflect these intrinsic organizational principles of PFC circuitry rather than task learning effects. It should be mentioned that these dorsal and ventral pathways are foundational features of primate cortical architecture, in contrast to the different organization of rodent cortex, where inter-areal interconnectivity is widespread. 82 On the other hand, FD changes in post-training compared to the corresponding intervals in pre-training were suggested to reflect working memory maintenance (first delay), while changes during the second delay suggest contributions of decision-making and rule learning since these interpretations were observed and proven by classic dorsal-ventral dissociations, 19 , 20 , 21 and with human neuroimaging evidence for separate spatial vs. object working memory systems. 52 Additionally, comparisons between dorsal and ventral regions within specific task and comparisons between the spatial and feature tasks within each recording region revealed differences in FD measures across conditions, particularly during the pre-training phase. These observations are discussed in relation to prior neurophysiological studies reporting dorsal-ventral functional distinctions. 48 , 49 , 50 It is worth noting that the simulation results highlight the sensitivity of FD methods to variations in firing rate and correlation structure in spike trains, supporting their applicability to neural recordings. More specifically, the simulation study demonstrated that KFD is highly sensitive to firing rate changes, HFD captures temporal correlations and multi-scale fluctuations, and BCFD highly emphasizes temporal correlations and global predictability. Together, they provide a multidimensional view of complexity that bridges firing-rate metrics and population decoding by directly quantifying how spike trains fill time with irregular or structured activity. Another outcome of our simulations is that the absolute direction of FD changes cannot be universally mapped onto cognitive processes. For example, KFD decreases with higher firing rates due to reduced ISI amplitude, while Higuchi and BCFDs decrease with stronger temporal correlations as spike trains become more predictable. What is robust across methods, however, is that FD changes indicate altered network dynamics (whether toward greater irregularity and multiscale variability [higher FD] or toward more predictable, low-dimensional patterns [lower FD]). In this sense, FD should be regarded as a biomarker of altered neural complexity, not a unidirectional marker of “more” or “less” cognition. Limitations of the study The presented results demonstrate that FD could emerge as a useful biomarker of altered neural complexity in prefrontal activity, capable of capturing both intrinsic prefrontal selectivity (before training) and training-related cognitive modulations (after learning). In addition, they provide a descriptive, method-dependent characterization of temporal complexity in neural spike-timing activity, and it does not directly index specific cognitive functions or task-specific neural coding. Different FD estimators are sensitive to distinct statistical properties of spike trains, and therefore changes in FD magnitude or direction cannot be interpreted uniformly across methods. In addition, the simulation analyses were designed to illustrate general estimator sensitivities under simplified and controlled conditions and do not capture the full complexity of real neural recordings. Experimental spike trains are shaped by multiple interacting factors, including population-level interactions, nonstationarity, heterogeneous firing dynamics, and contextual and temporal features intrinsic to trial structure (e.g., trial onset, fixation periods, and the sequential presentation of stimuli). Accordingly, significant FD modulation in certain conditions is interpreted as reflecting changes in neural complexity, while the absence of FD differences in other conditions does not imply a lack of neural or functional engagement. More generally, some aspects of cognitive processing may be supported by neural mechanisms, including oscillatory dynamics, phase relationships, or population-level coordination, that are not necessarily captured by complexity-based measures alone. Accordingly, FD should be viewed as a complementary measure of neural dynamics rather than as a mechanistic analyzer or a direct indicator of cognitive representations. In this study FD was computed using pooled ISI data within task epochs to ensure estimator stability with the available dataset. While pooling improves the reliability of FD estimates by providing sufficiently long spike sequences, it reduces single-neuron variability and excluded condition-specific analyses, such as separating match and non-match trials, particularly during the pre-training phase where differences in trial structure and stimulus presentations may differentially influence neural dynamics. As a result, the present findings reflect condition-dependent variations in temporal structure rather than population-coding or decoding mechanisms. Future studies with denser recordings and richer datasets will be essential to overcome these limitations. Condition-resolved analyses, including separate examination of match and non-match trials, error trials, and early learning phases, will allow a more precise assessment of how FD evolves with learning and task demands. Moreover, integrating FD with time-resolved population-level approaches—such as dimensionality reduction, state-space modeling, and decoding analyses—will be necessary to relate local spike-timing complexity to ensemble-level neural dynamics underlying working memory and decision-making. Overall, despite the aforementioned limitations, the present study applied three complementary FD methods (Higuchi, Katz, and box-counting) to analyze the complexity of ISI signals recorded from the DLPFC and VLPFC prefrontal cortices of macaque monkeys during spatial and feature working memory tasks. By examining both pre-training (passive viewing) and post-training (proficient performance) phases, our results demonstrate that FD can serve as a sensitive biomarker of neural complexity, revealing distinct functional patterns between these regions and across cognitive states. At the methodological level, the three FD estimators captured complementary aspects of spike-train dynamics: KFD was primarily sensitive to firing-rate-related irregularities, HFD reflected temporal correlations and multiscale fluctuations, and BCFD emphasized temporal correlations and global predictability. These distinctions clarify that different FD methods can yield complementary insights rather than redundant ones. Moreover, FD variations reflect shifts in neural dynamics (from irregular, high-dimensional activity to more predictable, low-dimensional regimes) providing a principled framework for linking local spike-time complexity with the adaptive coding capacity of neural networks. Functionally, our findings indicate that prefrontal complexity is modulated differently across regions, task types, and learning stages. Before training, both DLPFC and VLPFC displayed significant FD differences across task epochs, consistent with intrinsic dorsal-ventral specializations that persist even under passive viewing. Although both regions showed complexity change for both task types, the DLPFC exhibited greater sensitivity to spatial changes, whereas the VLPFC responded more to feature or object variations (patterns that align with established anatomical projections from dorsal and ventral visual streams). After training, complexity differences between pre- and post-training phases were pronounced during delay intervals, suggesting that the temporal structure of prefrontal activity evolves as working memory and rule-based decision processes become consolidated. These changes, observed in conditions where monkeys performed the task proficiently, support the interpretation that FD captures meaningful alterations in neural dynamics associated with cognitive control and learned rule application. Beyond reproducing known dorsal-ventral functional distinctions, this study highlights how fractal analysis enriches our understanding of prefrontal computation. By quantifying the multiscale irregularity of neural firing, FD provides a bridge between traditional rate-based metrics and higher-order population dynamics, offering a quantitative index of the system’s dynamical richness. In summary, fractal analysis of neuronal activity reveals that both the organization and adaptability of prefrontal dynamics can be quantitatively characterized by changes in neural complexity. The present findings establish FD as a robust and interpretable tool for studying the interplay between intrinsic circuitry and task-related cognition, offering a foundation for future research integrating fractal metrics with population coding and dynamic state-space analyses to further elucidate the mechanisms of working memory and decision-making in the brain. Resource availability Lead contact Requests for further information and resources should be directed to and will be fulfilled by the lead contact, Dr. Mohammad Reza Daliri ( [email protected] ). Materials availability This study did not generate new unique reagents or materials. Data and code availability • This article analyzes existing, publicly available data from the CRCNS repository (prefrontal cortex [PFC]; pfc-3), accessible at https://crcns.org/data-sets/pfc/pfc-3 . • This article does not report original code. • Any additional information required to reanalyze the data reported in this article is available from the lead contact upon request. Acknowledgments The authors acknowledge the contributors of the CRCNS pfc-3 dataset for making their recorded signals publicly available. We also thank CRCNS for providing access to these resources, which enabled the analyses presented in this study. The authors also declare that this work did not receive any funding. Author contributions Conceptualization, N.Z.-M., A.F., and M.R.D.; methodology, N.Z.-M., A.F., and M.R.D.; investigation, N.Z.-M. and A.F.; validation, N.Z.-M. and A.F.; writing – original draft, N.Z.-M. and A.F.; writing – review and editing, N.Z.-M., A.F., and M.R.D.; visualization, N.Z.-M., A.F., and M.R.D.; supervision, M.R.D. All authors interpreted the results, reviewed, edited, and approved the final version of the manuscript. Declaration of interests The authors declare no competing interests. Declaration of generative AI and AI-assisted technologies in the writing process During the preparation of this work, the authors used Grammarly, including its AI-assisted language editing features, to improve the clarity and quality of the English language in some sentences. The authors subsequently reviewed and edited all content and take full responsibility for the content of the publication. STAR★Methods Key resources table REAGENT or RESOURCE SOURCE IDENTIFIER Deposited data pfc-3 dataset CRCNS https://crcns.org/data-sets/pfc/pfc-3 Software and algorithms GraphPad Prism 10.6.1 GraphPad Software https://www.graphpad.com/ MATLAB 2023b MathWorks https://www.mathworks.com TISEAN 3.0.0 TISEAN https://www.pks.mpg.de/tisean/Tisean_3.0.0 FRACTALDIM (Leibovich and Tóth) EconPapers RePEc BOCB code t741507; https://econpapers.repec.org/software/bocbocode/t741507.htm Open in a new tab Experimental model and study participant details In this study, we analyzed the publicly available data (obtained from the CRCNS pfc-3 dataset) of four male rhesus monkeys (Macaca mulatta) conducting match-to-sample working memory tasks with no prior experimental experience. 83 , 84 , 85 All animal procedures in the original experiments were performed in accordance with National Institutes of Health guidelines and approved by the Wake Forest University Institutional Animal Care and Use Committee (Winston-Salem, NC), as reported by the data providers. 81 , 83 , 84 No additional ethical approval number applies to the present study, as no new animal work, data collection, or interventions were performed. It should be mentioned that the descriptions of the experimental paradigm, task structure, and recording procedures provided below are included for completeness and are based on the documentation and publications of the original data providers. During the experiment, the monkeys sat in a primate chair and their head was fixed while viewing the computer monitor positioned 68 cm away in a dark environment. An eye-tracking system was employed to monitor the eye position of the monkeys. A cylindrical chamber was implanted over the PFC of the monkeys. Neural activity was recorded by inserting up to 8 microelectrodes into the DLPFC and VLPFC (caudal half of the principal sulcus). 83 , 86 During each recording session, multiple records were performed as follows: every time the microarray was inserted into the brain, the electrodes in the vicinity of the neurons recorded the extracellular potential of the area. After recording the signal, action potential waveforms were sorted into distinct units through an automated clustering method utilizing the KlustaKwik algorithm. Only correct trials in every experiment were stored in the database, as specified by the data owners. 83 , 84 Recordings were performed after the animals had reached stable, high-level performance (∼85–90% correct), meaning that the post-training data reflect proficient task execution rather than early learning stages. In the pre-training phase, monkeys were only required to fixate while stimuli (spatial locations or features) were presented, without engaging in the match/non-match rule. In contrast, the post-training phase implemented the full working memory task: monkeys observed a sample stimulus, maintained it over a delay, and then categorized the test stimulus as match or non-match with a saccadic response. This design enables a direct comparison between passive stimulus-driven responses (pre-training) and cognitively demanding rule-based performance (post-training), while ensuring that all analyzed trials were correctly executed. The details of the experiment through two phases of Pre-training and Post-training stages are as follows. Pre-training stage In this experimental task, monkeys were required to maintain fixation on a 0.2° white square throughout the trials to receive a liquid reward. Visual stimuli were presented on the screen while the animals kept their fixation. Two types of stimuli were used: a spatial set and a feature set. The spatial set consists of a 2° square that appeared randomly in one of nine locations on a 3 × 3 grid with dimensions of 10°. To prevent anticipation, the position of the stimuli varied randomly across trials. After a one-second fixation on the initial point, the first stimulus was displayed for 500 ms. This stimulus then disappeared, followed by a 1.5-second delay, after which the second stimulus was presented for 500 ms. The location of the second stimulus could either match the first (referred to as a “match”) or be diametrically opposite (referred to as a “non-match”). Following another 1.5-s delay, the fixation point was extinguished, and the monkey was rewarded for successfully maintaining fixation throughout the task. 84 In contrast, the feature set of stimuli included eight white geometric shapes, each calibrated for size and luminance to fit within a 2-degree outline. Here, the blocks of trials were typically presented at the same spatial location, although each of the feature stimuli varied randomly (among the eight geometrical shapes) from trial to trial. The timing and duration of stimulus presentations were identical to the spatial set, which could either be the same (match) or different from the first feature stimulus (non-match). It must be mentioned that the feature set trials were typically presented after the spatial set trials, and stimuli were presented at the locations that evoked the best response during the spatial set. Post-training stage In the post-training phase, the monkeys are expected to determine whether the location of the second stimulus matches that of the first. The procedure of the feature and spatial tasks in these experiments is the same as in the pre-training stage. However, to indicate a match or mismatch, a green and a blue target (choice targets) appeared on the screen after a second delay period, allowing the monkeys to perform a saccade toward the green target for the match and the blue target for the mismatch. The choice targets appeared perpendicular to the direction of the stimuli, and the locations of the blue and green targets changed randomly across trials in each experiment. The correct selection ended with a liquid reward. 84 It is worth noting that only two out of the four monkeys performed the post-training task. Since the monkeys had to remember the first stimulus in order to determine the matching of the stimuli, we refer to the first delay period as the working memory duration. In contrast, during the second delay period, decision-making plays a more significant role, as the monkey must determine whether to saccade toward the green or blue target. More details of the experimental setup are explained in previous studies by Qi et al., and Kobak et al. 84 , 87 The results of the previous studies using this dataset demonstrated a clear dissociation between decision-making and working memory, both in terms of cognitive abilities and brain structure. 84 , 87 , 88 Therefore, in this study, two delay intervals were investigated separately relative to the fixation interval in order to provide a clear comprehension of the differences. Method details Fractal features Fractal geometry has recently attracted the interest of numerous researchers due to its remarkable properties and broad applications. The FD is particularly valuable for analyzing complex natural systems, as many phenomena in nature cannot be adequately described by integer dimensions. 89 , 90 As a fundamental concept in fractal geometry, FD quantifies how the complexity of a pattern changes across scales. In other words, FD is a valuable metric that quantifies the change in detail within a fractal as the scale varies. This scaling can occur in either space or time. Following the introduction of the Higuchi method for FD computation by Higuchi in 1988, 91 Katz’s method was subsequently employed as a different technique to analyze electroencephalogram (EEG) during sleep and awake states. 92 As a result, the investigation of the fractal characteristics as potential biomarkers in brain activities was initiated. Numerous studies have focused on differentiating between individuals with normal cognitive abilities and those with cognitive impairment or disease. 79 , 93 , 94 , 95 Indeed, FD methods have been successfully applied to non-invasive neural recordings such as EEG and fMRI, where they have been associated with cognitive state and clinical conditions. For example, EEG FD measures (e.g., HFD, KFD) have revealed altered brain dynamics in epilepsy, 96 Alzheimer’s disease, 97 , 98 and schizophrenia, 99 and have also been linked to attentional load, vigilance, and depth of anesthesia. 100 , 101 Furthermore, Goh et al., 102 examined various FDs and nonlinear indices to distinguish between the EEG signals of individuals with normal cognitive function and those with dementia. The FDs of Higuchi and Katz exhibit notable disparities when comparing the average and encephalopathy data, and Higuchi’s FD has exceptional discrimination ability to distinguish between the two groups effectively. 103 Recent reviews further highlight the promise of FD and related complexity measures in EEG and psychophysiology as biomarkers of neural integrity and cognition. 104 Similarly, FD analyses of fMRI blood-oxygen-level-dependent (BOLD) signals (scaling-based methods related to Higuchi/box-counting) have distinguished resting from task-engaged networks, 105 , 106 and captured age- and disease-related change. 107 Another area of research focuses on examining various brain states, and executive functions, or predicting the onset or occurrence of a particular event. 108 , 109 , 110 A more detailed investigation has been done in other studies through invasive recording techniques focusing on local field potentials and single-neuron activity to explore the intricacies of neural dynamics. 72 , 111 , 112 , 113 The human brain is recognized as a complex and dynamic system due to its intricate nature. The increase in complexity of brain activity and the movement toward a critical threshold between order and disorder plays a role in changing brain functionality, leading to the emergence of complex and dynamic patterns of neural activity. 66 , 114 , 115 Also, cortical complexity can vary between individuals based on age, experience, and different pathological conditions that impact the brain. 116 Research on cognitive functions, such as working memory, has revealed that without changes in firing rate, the nonlinear dimensionality of neurons in the middle temporal cortex increases when the spatial working memory matches the neurons’ receptive field. 53 Identifying top-down modulation in cortical firing patterns was successfully achieved by confirming the HFD as the most robust indicator of working memory content. 117 Multiple methods exist for calculating the FD, among which the Higuchi, Katz, and box-counting methods have gained more interest and have been used in most researches. 56 , 118 , 119 , 120 This research employed Higuchi, Katz, and box-counting techniques to compute the FD. Higuchi and Katz’s methods were utilized as the two initial approaches for calculating the FD. Alternatively, the box-counting technique can be interpreted as the count of elements that the fractal encompasses on a grid, which increases as the number of elements grows. These three methods are sensitive to different characteristics of the signals, which do not always converge on the same result, and thus reporting method-specific findings is meaningful rather than redundant. It should be mentioned that due to these changes in the sensitivity of these FD methods, multiple FD methods are commonly used together in analyzing the phenomena. 103 , 119 , 121 , 122 In this study, the Higuchi and Katz fractal dimensions were computed using implementations from the TISEAN software package (version 3.0.0; Nonlinear Time Series Analysis), adapted for analysis of inter-spike interval time series. 123 In addition, the box-counting fractal dimension was estimated using the FRACTALDIM MATLAB function, based on the algorithm described by Liebovitch and Tóth. 124 All FD analyses were performed in MATLAB (MathWorks). In the following, the algorithms for calculating the FD and their characteristics are described. Higuchi algorithm The Higuchi method estimates the FD of a time series using a systematic procedure. It begins with a discrete time series X = { x (1), x (2), …, x ( N )}, where each x represents a data point and N denotes the total number of points. For each integer k (where k ≥ 2), the method creates k new time series x m k defined as x m k = { x ( m ) , x ( m + k ) , … , x ( m + N − m k k ) } , m = 1 , … . k (Equation 1) Here, m varies from 1 to k , indicating the starting point, while k serves as the time interval. For instance, if k = 20 for a time series with length N = 1000, the resulting new time series will include x 1 20 = [ x ( 1 ) , x ( 21 ) , x ( 41 ) , . . . , x ( 981 ) ] ; x 2 20 = [ x ( 2 ) , x ( 22 ) , x ( 42 ) , . . . , x ( 982 ) ] ; ⋮ a n d s o o n , u p t o x 20 20 = [ x ( 20 ) , x ( 40 ) , x ( 60 ) , . . . , x ( 1000 ) ] : (Equation 2) For each new series x m k , the length L m ( k ) is computed using the formula: L m ( k ) = 1 k [ N − 1 k N − m k ∑ i = 1 N − m k | x ( m + i k ) − x ( m + ( i − 1 ) k ) | ] (Equation 3) The average length L ( k ) for the k sets is then calculated as: L ( k ) = 1 k ∑ m = 1 k L m ( k ) (Equation 4) Finally, a plot of ln ( L ( k )) against ln (1/ k ) is created, where the slope of the best-fit line through these points corresponds to the Higuchi FD. This dimension typically ranges from 1 to 2 for one-dimensional data, reflecting the degree of fractal behavior. As an example, Figure S1 A illustrates a non-stationary signal segmented into subseries for k = 2,3,4,5. When k = 2, two complementary subseries are generated: one starting from the first point and capturing every other sample, and another starting from the second point and covering the remaining samples. Figure S1 B plots log 10 L ( k ) versus log 10 k , and the slope of the ordinary least squares (OLS) fit provides the HFD. This method emphasizes the multi-scale temporal roughness of the signal. The Higuchi method facilitates the analysis of time series data in its original domain, eliminating the need for phase space reconstruction. 91 It is sensitive to frequency content and to temporal correlations and multi-scale structure (e.g., long-range dependencies, oscillatory modulation, and bursting), 91 , 125 and is less sensitive than KFD to pure amplitude/sampling changes. 122 , 126 HFD is often among the most accurate estimators of the “true” FD on synthetic benchmarks and EEG data, though computationally more demanding than simple length-based indices. 121 It serves as a valuable tool for measuring complexity across various fields, including image analysis (such as fMRI data 127 , 128 , 129 , 130 , 131 ), and biological signals (e.g. EEG recordings 64 , 132 ). Katz algorithm Katz’s method computes FD as the logarithm of the curve length relative to maximum displacement using the following relations. 92 If the signal in a window with n samples is considered as a set of consecutive points constituting the curves with the polyline length (L) (sum of distances between successive points), its FD, can be obtained from 5 : F D k = log ( L ) log ( d ) (Equation 5) where L is the total length of the curve and d is its diameter. For a time series, the entire curve length equals the sum of the Euclidean distance between successive points. The diameter d is the maximum distance between the initial starting point and the farthest point. Equation 5 must be corrected by an average step “ a ,” which is the average distance between consecutive points. In this case, by replacing L with L / a , we get to 5 as follows: F D k = log ( L a ) log ( d a ) = log n log n + log ( d L ) (Equation 6) As an example, Figure S2 A shows windows of n = 10,20,30, and 40 points. For each window, L , d , and a are computed. Figure S2 B plots log (L/a) versus log (d/a); the fitted slope approximates the average Katz FD (KFD). This reflects how signal complexity scales with observation length. KFD is particularly sensitive to local irregularity and amplitude variation, 92 , 122 , 126 and compared with HFD, its estimates can be driven more by amplitude/record-length/sampling than by intrinsic multi-scale correlations. 122 , 126 While it often underestimates absolute FD for long or self-affine signals, 122 , 126 it is computationally efficient, 92 and performs reliably for short windows in discrimination tasks. 121 Box-counting algorithm The Box-counting fractal dimension (BCFD) is one of the FDs that due to its relatively straightforward mathematical calculations and experimental estimates, has found widespread application across various fields. Box-counting method studies the effect of curved coverage with a set of area elements. The area value of elements with given sizes is counted to determine how many boxes are needed to cover the curve completely. Finally, when the size of the elements tends to zero, the total area covered by the area of the elements will converge to the dimension of the curve. Therefore, the following formula is used to calculate the FD: D B = lim ε → 0 log N ( ε ) log ( 1 ε ) (Equation 7) In the above equation, ε equals the size of the boxes, and N ( ε ) represents the number of boxes used to cover the curve with specific sizes. Since the primary method has a high computational order, other methods have been proposed to reduce the amount of memory. As an example, Figure S3 A illustrates grid partitions of size 2 × 2, 4 × 4, 8 × 8, and 16 × 16; at each step, the occupied boxes are counted. Figure S3 B plots log 10 N(ε) versus log 10 (1/ε); the slope of the fit yields the BCFD, reflecting how densely the trajectory fills space across scales. The BCFD is the most classical method: it covers the signal (or image) with grids of boxes across multiple scales, quantifying global geometric complexity. 61 Its estimates are sensitive to choices such as resolution, scale range, and even grid origin. 133 , 134 On finite, noisy, or short biological time series it can underestimate the complexity between-condition differences (estimates become less discriminative). 122 , 135 It is better suited for large datasets or images (e.g., MRI/CT, ERP/BCI image-based tasks) 119 , 136 , 137 , 138 ; although, runtime grows with resolution and box ranges, and many works propose optimizations for that reason. 119 , 139 A fast method has been introduced by Leibovich and Toth. 140 In this method, there is no need to fully count each cell in high resolution. With statistical data sampling, the complexity of calculations is reduced from the number of boxes in the network to the number of points. Simulated Poisson spike trains To show the distinct sensitivities of the three methods in ISIs derived from neural data (as being done in this paper on real neural data), we performed a simulation using well-known homogeneous Poisson model of stochastic neuronal firing to generate spike trains. 80 The Poisson model occurs very naturally in situations in which a large number of independent components all have the capacity for generating an event, but none of them are very likely to do so at a given time. For example, it does an excellent job of describing radioactive decay. Early neurophysiologists were struck by the irregularity spike trains recorded from neurons in vivo and thus the Poisson model of neural firing was born. Even these early neurophysiologists were aware that the firing rates of real neurons are not stationary and don’t have strictly independent increments. Yet, despite these variations of opinions, the Poisson model does a remarkable job of describing firing variability of cortical neurons. In our simulation, spike trains were generated with firing rates systematically varied between 4 and 7 Hz to investigate the effect of firing rate on the FDs for 1.5 s time length, the same as the time length in delay intervals in the considered task (500 trials × 30 repetitions per each condition). It should be mentioned that the firing rate and time interval length ranges are chosen based on empirical values from our dataset. ISIs were computed from the simulated spike trains, pooled, and analyzed with the three fractal methods similar to the analysis performed in this paper for the real data. In investigation of temporal correlation, exponential ISIs were generated as a weighted combination of the previous ISI and a newly drawn exponential ISI with three correlation levels (ρ = 0, 0.3, 0.6), thereby preserving the mean firing rate but introducing first-order serial correlations between successive intervals. This approach breaks the memoryless property of the Poisson process and allows us to examine how FD estimators respond to temporal dependencies beyond simple rate irregularity. Data analysis The employed dataset consists of the data from four monkeys, namely ADR (Pre & Post Training), ELV (Pre & Post Training), BEN (Pre-Training), and SCR (Pre Training, specifically focusing on the Dorsal Region), available online at the Collaborative Research in Computational Neuroscience (CRCNS) database. This dataset can be categorized into three distinct groups based on different criteria. Firstly, the data can be categorized as either pre-training or post-training, indicating the stage at which the observations were made. Secondly, the data can be classified according to the region of interest, specifically whether it belongs to the dorsal or ventral region. Lastly, the data can be separated based on the type of task performed (spatial or feature tasks). By organizing the data into these three groups, we can gain valuable insights from different aspects and analyze the results more effectively. Subsequently, the analysis is conducted within three predetermined time intervals: the fixation phase, the initial delay (‘Delay 1’) following the first stimulation, and the subsequent delay (‘Delay 2’) following the secondary stimulation. In the “Fixation phase”, each neuron’s initial level of activity was established during pre- and post-training trials. These trials required the monkey to focus on the fixation point without any cue. In both delay intervals, 300 ms after the visual cue offset has been excluded from the signal, thereby isolating the memory-related activities. Furthermore, the ISI is computed from the spikes of individual neurons, and these intervals are organized into distinct time series and sorted according to the trial number. More specifically, within each recording session, we accessed the spike-sorted single-unit trials separately for the two regions of interest (DLPFC and VLPFC) and task type (spatial or feature). Task epochs (fixation, delay1, delay2) were extracted using the onset times provided in the dataset after the fixation period. Within each condition (region × task × epoch), ISIs were first computed for the trials of each neuron. The ISIs of neurons recorded on the same electrode were then concatenated according to their temporal order. Pooled ISIs from different electrodes were subsequently combined within each session. Because the available dataset included only correct trials and did not contain a sufficient number of separately labeled match and non-match trials for all sessions, we analyzed pooled ISI sequences within each task condition. In the next step, the FD is computed from these time series. All analyses were performed separately for each monkey. Within each condition, FD values were first cleaned for each monkey separately by removing NaN values, and zeros to ensure biologically valid spikes. Robust outlier trimming was then applied using the median absolute deviation (MAD): for each session, the robust Z score of each FD relative to the median was calculated, and extreme outliers (|z| > 3.5) were excluded. 141 , 142 Normality of the cleaned distributions was assessed with the Kolmogorov–Smirnov (K-S) test. 143 The cleaned FDs of neurons recorded on the same electrode were concatenated in temporal order, and pooled FDs from different monkeys were combined to create a continuous session-level sequence. To combine FDs across monkeys, pooled (weighted) means and pooled sample variances were calculated, 144 , 145 preserving both within- and between-session dispersion. The whole analyzing procedure is depicted in Figure S4 , representing the schematic of the steps we undertook. It is worth noting that one of the important indexes for checking the predictability and stability of a system can be assessed by the Lyapunov exponents (LEs), which quantify the exponential separation rate over time between two initially neighboring points. 146 Positive and negative LE values indicate whether the state trajectories diverge or converge exponentially. This divergence and convergence can be visually represented as the stretching and folding of the phase space, respectively. Meanwhile, a system, based on its dimension, has multiple LEs. The largest LE value primarily determines the predictability of the system. Higher values of the largest LE indicate a more complex signal. In signals, only the largest LE can be investigated. In this experiment, the median of the largest LE was positive for the signals of the considered dataset, meaning that the FD is meaningful in fractal theory. Quantification and statistical analysis Statistical analyses and graphical visualizations were performed using GraphPad Prism (version 10.6.1, GraphPad Software). Additional statistical details and complementary computations (effect sizes and powers) were carried out using MATLAB (MathWorks). Neuronal spike trains were converted to inter-spike interval (ISI) time series, and fractal dimensions (FDs) were computed using the Higuchi, Katz, and box-counting methods. For each neuron, FDs were calculated separately for each task condition (spatial or feature), cortical region (dorsolateral or ventrolateral prefrontal cortex), training phase (pre-training or post-training), and task interval (fixation, delay 1, and delay 2). To reduce the influence of outliers, FD values were averaged across trials within each condition prior to group-level statistical analysis. Statistical comparisons were performed using two-way analysis of variance (ANOVA). More specifically, a two-way ANOVA was conducted with time interval (fixation, delay 1, and delay 2) as one factor and task-region–training condition as the second factor, comprising spatial and feature tasks recorded from dorsal and ventral prefrontal cortex during pre-training and post-training phases (eight conditions). When significant main effects or interactions were detected, post hoc pairwise comparisons were conducted using the Tukey–Kramer correction for multiple comparisons. For the simulation study examining firing rate effects, statistical comparisons were performed using one-way ANOVA, with firing rate as the main factor. When significant effects were observed, post hoc comparisons were conducted using the Tukey–Kramer correction. To assess the combined effects of firing rate and correlation strength, a two-way ANOVA was conducted with firing rate and correlation level as fixed factors, and interaction effects were evaluated. When significant main effects or interactions were detected, post hoc pairwise comparisons were performed using the Tukey-Kramer correction. In addition, simple-effect analyses were conducted to examine firing rate differences within each correlation level and correlation differences within each firing rate. Effect sizes were quantified using Hedges’ g , and statistical power was estimated for significant effects. Normality assumptions were assessed using Kolmogorov-Smirnov test and were deemed acceptable for ANOVA-based analyses. Statistical significance was defined as p < 0.05. Statistical significance is indicated in the figures using asterisks. The corresponding p -value ranges represented by the asterisks (ns = not significant, ∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001, and ∗∗∗∗ p < 0.0001), as well as the values represented by the error bars, are defined in the figure legends. 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[ Google Scholar ] Associated Data This section collects any data citations, data availability statements, or supplementary materials included in this article. Supplementary Materials Document S1. Figures S1–S4 mmc1.pdf (947.7KB, pdf) Data Availability Statement • This article analyzes existing, publicly available data from the CRCNS repository (prefrontal cortex [PFC]; pfc-3), accessible at https://crcns.org/data-sets/pfc/pfc-3 . • This article does not report original code. • Any additional information required to reanalyze the data reported in this article is available from the lead contact upon request. 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