The authors have declared that no competing interests exist.
Conceived and designed the experiments: KMK FF. Performed the experiments: KMK. Analyzed the data: KMK. Wrote the paper: KMK FF.
The importance of the large number of thin-diameter and unmyelinated axons that connect different cortical areas is unknown. The pronounced propagation delays in these axons may prevent synchronization of cortical networks and therefore hinder efficient information integration and processing. Yet, such global information integration across cortical areas is vital for higher cognitive function. We hypothesized that delays in communication between cortical areas can disrupt synchronization and therefore enhance the set of activity trajectories and computations interconnected networks can perform. To evaluate this hypothesis, we studied the effect of long-range cortical projections with propagation delays in interconnected large-scale cortical networks that exhibited spontaneous rhythmic activity. Long-range connections with delays caused the emergence of metastable, spatio-temporally distinct activity states between which the networks spontaneously transitioned. Interestingly, the observed activity patterns correspond to macroscopic network dynamics such as globally synchronized activity, propagating wave fronts, and spiral waves that have been previously observed in neurophysiological recordings from humans and animal models. Transient perturbations with simulated transcranial alternating current stimulation (tACS) confirmed the multistability of the interconnected networks by switching the networks between these metastable states. Our model thus proposes that slower long-range connections enrich the landscape of activity states and represent a parsimonious mechanism for the emergence of multistability in cortical networks. These results further provide a mechanistic link between the known deficits in connectivity and cortical state dynamics in neuropsychiatric illnesses such as schizophrenia and autism, as well as suggest non-invasive brain stimulation as an effective treatment for these illnesses.
The brain mediates behavior by orchestrating the activity of billions of neurons that communicate with each other through electric impulses. The transmission of these action potentials is surprisingly slow for a large fraction of these connections. Given the importance of precise timing of neuronal activity, the function of these slow connections has remained a puzzle. We here used computer simulations to investigate how slow connection speeds alter the overall activity patterns of two brain networks. We found that these connections enable the interconnected networks to generate distinct activity patterns such as different types of waves of electric activity. Our results therefore suggest that the slow transmission of electric impulses in the brain is not a “design flaw” but rather plays an important role in enabling the brain to generate a richer set of activity patterns. The ability of the brain to switch between different activity states is crucial to normal cognition, and abnormalities in switching behavior are associated with cognitive symptoms in psychiatric disorders such as schizophrenia and autism. It is therefore promising that we were able to control transitions between different activity states with non-invasive brain stimulation in our simulations, suggesting a novel approach to the treatment of these illnesses.
Cognition emerges from the organized temporal structure of electric activity in large, interconnected cortical networks
Systematic parameterization of network topology in computational models has demonstrated that connections between random pairs of distant, excitatory neurons within a network enhance temporal synchronization, whereas predominantly local connectivity between neighboring excitatory neurons facilitates macroscopic activity patterns such as oscillations and planar and spiral waves that propagate through the network
Mathematical studies of the effects of delays on coupled oscillators have predicted diverse results as a consequence of delays. Foundational papers have found that delays between coupled systems produce stability under certain parameters
We hypothesized that slower long-range projections may enrich overall network activity by counteracting and disrupting the intrinsic, spontaneous dynamics of individual networks. According to our hypothesis, slower projections provide perturbations that are ill-timed to synchronize networks and therefore enable different activity trajectories that individual networks are unable to generate. To test this hypothesis, we used large-scale computer simulations to ask what role long-range projections with propagation delays may play in organizing the overall dynamics of two interconnected cortical networks with intrinsic spontaneous dynamics similar to isolated cortical networks
To understand the effect of long-range projections (LRPs) on the dynamics of two interconnected cortical networks, we built a large-scale computational model of two networks connected by LRPs (
(A) Network model. Each network consists of 160,000 excitatory pyramidal neurons (PYs) and 40,000 inhibitory interneurons (INs). Synaptic connectivity: PY-PY, IN-PY: AMPA synapses; IN-PY: GABAA synapses. PYs in both networks are mutually connected by AMPAergic long-range projections (LRPs). (B) PY activity in each network. Left: No LRPs. Right: With LRPs. (C) Time snapshots of binned PY firing rates. Without (top) and with LRPs (bottom), Network 2 fires before Network 1; with LRPs, UP states are synchronized. Onset site remains the same (red arrow). Color represents instantaneous firing rate. (D) Phase space plots comparing the percentage of PYs firing in Network 1 with the percentage firing in Network 2. Trajectory close to unity line indicates synchronization of PY activity across networks in the presence of LRPs (right). (E) Correlation coefficients between homologous PYs across networks. Area of reduced correlation coefficients corresponds to the region of initiation of UP states in Network 2 (arrow).
Both with and without LRPs, UP states emerged as initially localized “regions of initiation” that then expanded through the local excitatory connectivity (circular patterns in
To mimic realistic delays in action potential propagation along low-diameter and unmyelinated fibers that connect different networks, we next added physiologically plausible delays
We clustered the simulation outputs with linkage analysis using the peak cross-correlation value, which measures the overall synchronization of the two PY networks (dendrograms in
Simulations clustered by maximum cross-correlation value with linkage analysis. Phase space plots and cross-correlograms shown for all clusters (defined by 90% of full tree). Dark blue represents clusters with the greatest magnitude of cross-correlation maxima, followed by cyan, green, and black. (A) LRP delay: 0 msec. (B) LRP delay: 50 msec. Insets: Phase space plots and cross-correlograms.
We then examined how these different synchronization patterns impacted the intrinsic dynamics within the individual networks. Indeed, inspection of the spatio-temporal activity profiles revealed the occurrence of three distinct patterns, which can be classified as network states. Typically, networks were in a rapid fire (RF) state, with most PYs in the network firing almost simultaneously and the network as a whole demonstrating slow oscillatory behavior (
(A–C) Cortical activity states characterized by different spatio-temporal activity patterns. Top: PY activity throughout simulation. Bottom: Time snapshots of PY activity. (A) Rapid fire state (RF): synchronized PY firing within a network. (B) Slow propagating state (SP): Activity originates in one or a few places and slowly traverses through the network. (C) Spiral wave state (SW): waves propagate from a central rotor in a spiral shape. (D) Percentage of time simulations spent in each state by delay, separated by network state (RF, SP, SW from left to right). (E) Frequency of state transitions by delay. All error bars represent s.e.m.
Next, we asked how the occurrence of these three different macroscopic network states depended on LRP delays. We found that most interconnected networks followed an RF pattern, especially for short LRP delays (
In order to further evaluate the robustness of this result, we also tested the effects of a distribution of delays. We ran two sets of simulations, the first with delays uniformly distributed ±20% of the mean and the second with delays uniformly distributed ±100% of the mean. Our results indicate that wider distributions resulted in fewer state transitions (
We then analyzed the transitions of individual simulations through these metastable spatio-temporal activity patterns over time (
(A) Example simulation with state transitions. Left to right: SP, SP/SW, RF, SW, SP, RF. (B) PY activity profile. Arrows correspond to time snapshots in (A). (C) Spectral power as a function of network state. Left: Time-averaged spectrum exhibits strong peak at endogenous network frequency (fmax≈3 Hz). Middle: Spectrogram shows pronounced changes in power at fmax (dashed box). Right: Time-course of power at fmax. Colored arrows correspond to time snapshots in (A). RF exhibited highest power at fmax.
To further understand these different network states, we next applied perturbations to probe the stability of each state. Specifically, we simulated transcranial alternating current stimulation (tACS), which has recently emerged as a promising treatment for psychiatric and neurological illnesses because of its hypothesized ability to selectively manipulate temporal structure of cortical network activity
We here used this stimulation modality to probe the dynamic properties of the different activity states that emerged from LRPs with propagation delays. We found that tACS at 3 Hz (close to the endogenous frequency of the individual networks) not only enhanced the synchronization between the two networks but switched the two networks to the fully synchronized, RF state (
Transcranial alternating current stimulation (tACS) induces outlasting changes in cortical state. (A) Example 1: PY activity plots and spectrograms of simulation receiving tACS. Red: tACS waveform. Network 1 was in RF at onset, which was enhanced by tACS with an outlasting effect on oscillation power. Network 2 began in SW, which was disrupted by tACS, and switched to RF that persisted after removal of tACS. (B) Example 2: Both networks began in RF but were disrupted by the onset of tACS. During tACS the networks exhibited SP with reduced power at 3 Hz compared to pre-onset behavior. After tACS, both networks switched to SW. (C) Percentage of time in each state by delay before, during, and after tACS. During tACS, the amount of spent in SP increased compared to before stimulation and was independent of delay. After tACS, time spent in SP was reduced compared to before tACS (for 10, 30, and 50 msec delays). There was also an increased amount of SW for all delays. (D) Transition probabilities between the network states without tACS (baseline), at the onset of tACS, and at the removal of tACS. Green numbers: Increase from baseline. Red numbers: Decrease from baseline. At the onset of tACS, SW transitions to either RF or SP, while SP and RF had a greater likelihood of transitioning to the other state. Once tACS was removed, SP is maintained less than before stimulation, with a greater chance of transitioning to both SW and RF. SW had a decreased chance of transitioning to SP.
Interestingly, a small fraction of the simulations did not show this enhancing effect of tACS. Rather, in these cases, tACS switched the networks from RF to either SW or SP states (
Given these distinct effects of the same stimulation protocol in different simulations, we determined the relative occurrence of the different states and the state transition probabilities for all simulations (including all propagation delays, fraction of LRPs, and strength of LRPs) as a function of tACS. In the control condition before onset of stimulation (
Overall, the state-dependent transition probabilities in the absence of tACS, at tACS onset, and at tACS removal (
We then compared how networks behaved together and found that in the absence of stimulation, both networks were in the RF state for the majority of simulations (
Having established that tACS affects the spatio-temporal activity of two interconnected networks, we next quantified the effect of tACS on the power of the network activity at the stimulation frequency (3 Hz). First, we looked at the effectiveness of tACS to entrain two networks during stimulation by comparing the power during stimulation to the power before stimulation. We found that tACS enhanced the power at 3 Hz of both PY networks during stimulation for most simulations, indicating its ability to entrain networks (
(A) Increase in oscillation power by tACS across the two networks. (B) Outlasting effect of stimulation across the two networks. (C) Change in power during tACS versus change in power after tACS for each network over all simulations. Change in power after tACS is correlated to the change in power during tACS, demonstrating an outlasting effect of stimulation. (D) Correlation coefficients by delay for each plot in A–C (significant for all delays,
To investigate how this outlasting effect of tACS related to the entrainment during stimulation, we compared the enhancement of power at 3 Hz during stimulation to the enhancement of power after stimulation (
Although tACS typically entrained networks to a 3 Hz RF state, occasionally it had an opposite effect by disrupting RF during tACS and causing it to enter SW after tACS. We examined these network dynamics to determine which factors influenced such disruption. Networks that ended in SW after tACS were most often in SP or SW during tACS and only very rarely in RF (
(A) Distribution of behavior during tACS for simulations that ended in SW. Most SW simulations were in SP during tACS. (B) Left: Mean PY activity for the first 2.5 seconds of all simulations, grouped by behavior during tACS. Simulations that entered SP or SW during tACS had similar behavior before tACS. Right: Mean PY activity at onset of tACS (t = 2.0). Simulations that entered SP and SW had greater activity at onset than simulations in RF. Error bars indicate s.e.m. (C) Influence of parameters on behavior during tACS. Left: LRP conductance and connectivity. Middle: LRP conductance and delay. Right: LRP connectivity and delay. Weak conductance and high LRP connectivity (low P(local)) predisposed a network against RF during tACS. This effect was enhanced with shorter delays. (D) Influence of parameters on behavior after tACS. Heat maps same as above. Weak LRP conductance and weak connectivity made a network more likely to enter SW after tACS, with no pronounced effect of delay.
When looking at PY activity before tACS, networks in RF during tACS had no specific pattern of activity while networks in SP or SW had a clear temporal structure in their PY activity prior to the onset of tACS (
Along with the above described network excitation, however, other factors also facilitated switching to a non-RF state during tACS. Higher LRP connectivity (i.e. lower P(local)) and lower LRP conductance (G(LRP)) both made networks more likely to enter a non-RF state, and these effects were increased with lower delays (
The relative prominence of SW after removal of tACS led us to measure the stability of the SW state. We first examined stability of SW in the absence of tACS and found that SW was a metastable state (
To further probe the mechanisms behind state disruption by tACS, we next simulated antiphase tACS using the same parameters but with the stimulation signal for the two networks phase-shifted by 180 degrees (
(A) Schematic of antiphase stimulation. (B) Top: Maximum cross-correlation value, indicating similarity of network behavior. Antiphase tACS disrupts network behavior. Bottom: Offset of maximum cross-correlation value indicating phase difference between two networks. Phase difference increased greatly during antiphase tACS but returned to near-baseline levels after removal. (C) Example of strong antiphase tACS behavior. Left: cross-correlogram. Right: PY activity. Both networks fire at 3 Hz during tACS but in antiphase. (D) Example of interspersed weak firing tACS behavior. Networks have strong out-of-phase peaks, but weaker peaks are in phase with the other network. (E) Example of breaking from RF behavior. Network 2 was disrupted by tACS and entered SW after tACS. (F) Effects of parameters on antiphase tACS behavior. Higher connectivity and conductances made interspersed weak firing more likely, while lower LRP connectivity and conductances increased the amount of strong antiphase and breaking from RF behavior. Delays only had a minor effect.
By examining the effects of parameters on behavior during antiphase tACS, the causes of RF disruption can be more thoroughly uncovered. Higher LRP connectivity (i.e. low P(local)) and higher LRP conductance made interspersed weak firing more likely (
Finally, an interesting behavior arose during antiphase stimulation where the two networks entered a high-frequency (>8 Hz) antiphase state (
We used simulations of two large, interconnected cortical networks to study how LRPs that connect the two networks affect the overall macroscopic dynamics. We found that introducing physiologically plausible delays to the LRPs greatly enhanced the repertoire of emergent dynamics, measured not only by synchronization between the two networks but also by the intrinsic spatio-temporal dynamics. Our results therefore suggest small-diameter and unmyelinated projection axons with propagation delays play an important role in enriching the landscape of cortical activity states. This finding contrasts with the traditionally assumed role of long-range connections to enable zero-lag synchrony between different cortical areas
Our study exclusively utilized computer simulations and therefore has the same caveats as any modeling study. First, the level of abstraction for the model requires consideration. We used computationally efficient, yet biologically plausible model neurons since we were interested in studying the effect of connectivity without confounding the results with the effects of conductance-based, Hodgkin-Huxley-style neuron models, which could model more sophisticated intrinsic cellular dynamics. A reduced model investigating the bifurcations involved in state transitions would provide further insight into network dynamics, although for this study it would reduce the applicability of our findings to the development of novel brain stimulation paradigms. Second, any biologically plausible finding in a computer simulation needs to withstand tests for reasonable robustness to parameter variations. The entire data set presented in this study was based on multiple runs of every simulation with different instantiations of the randomized variables (such as intrinsic excitability and target neurons for global random connections). Third, we believe that the value of most modeling studies can be readily assessed by the type of predictions they make that can then guide subsequent research, whether it be further computational work, wet lab bench studies, or even human preclinical trials. We therefore use the remainder of the discussion section to outline and discuss what we think are the implications and predictions of our results for the study of brain stimulation and network deficits in diseases with altered CNS connectivity such as schizophrenia, autism, and multiple sclerosis.
Brain stimulation, whether through implanted electrodes such as in deep brain stimulation
Our study suggests that rather than reorganizing synaptic strength, tACS can induce a switch between different macroscopic activity states that are part of a repertoire of cortical states mediated by LRPs with propagation delays. Interestingly, we also found that the same stimulation paradigm had the opposite effect in a (small) subset of simulations where the stimulation reduced the synchronization; these results demonstrate that (1) the ongoing network dynamics (i.e. network state) and the underlying network topology determine the response to brain stimulation and (2) a global stimulus does not necessarily enhance synchronization. Antiphase tACS, a stimulus designed to disrupt synchronization, caused a set of new behaviors during stimulation, but in most cases failed to create antiphase structure between the networks as an outlasting effect. Consequently, the outlasting effects of stimulation are dependent on the phase of stimulation as well as the intrinsic network structure.
As part of a computational model, conclusions drawn from our simulations of tACS are limited by the size of our networks and the fact that each PY receives the same magnitude of stimulation; however, simulated variance of tACS current amplitude has previously been found to have no effect on network response
Pathological changes in connectivity in the central nervous system (CNS) are a hallmark of many neurological and psychiatric illnesses. For example, schizophrenia is often called a connectivity disorder due to the findings of aberrations in white matter and lack of functional connectivity in both functional MRI and electroencephalogram (EEG) studies
We used computer simulations of large-scale, interconnected cortical networks in this study and found that long-range projections with physiological delays can play an unanticipated role in generating multistable network dynamics in cortex. Therefore, the so far neglected slow connecting fibers between cortical areas may not be a “flawed design” that prevents large-scale synchronization of cortical areas but rather enables the emergence of additional, qualitatively different network states that likely serve different neural computations. The ability of non-invasive brain stimulation to change these network states points to a promising treatment option for neuropsychiatric disorders involving abnormal connectivity and network dynamics.
We used the Izhikevich model
For PYs, parameters
Synapses were model by conductances that were updated with a step in case of a presynaptic action potential and that were subject to exponential decay otherwise. All synapses of a given type were lumped together into a single synapse to increase computational efficiency of the simulations
All simulations in this study consisted of two connected networks. Each network consisted of two layers, a PY network (400×400 model neurons arranged on a two-dimensional grid) and an IN network (200×200 model neurons arranged on a grid). The large number of neurons was motivated by the fact that tACS is likely to act as a global weak perturbation similar to the endogenous electric field
All cells received a current injection
The effect of the electric field resulting from tACS was modeled by injecting a small current into all PYs
Network activity profiles were determined by the fraction of PY neurons that were firing over time. Both normalized cross-correlations and spectrograms were based on these activity profiles by network. Spectrograms were computed by Wavelet transformation with Morlet wavelets (0.5 to 10 Hz in 0.5 Hz step-width). Macroscopic spatio-temporal activity states were distinguished by the median PY activity peaks (percent PYs firing) in 1 sec bins. Peaks (UP states) were extracted with the Matlab findpeaks function (threshold: 1% of maximum, dead time 50 msec, Mathworks, Natwick, MA). Rapid fire (RF) was assigned to peak values >60% of total number of PYs in the network, slow propagating (SP) was assigned to values 15–60%, and spiral wave (SW) was assigned to values <15%. Relative time spent in different states was determined over all simulations with the two networks considered together. State-dependent transition probabilities were determined for a 1 sec window before stimulation onset, 1 sec after stimulation onset, and last 1 sec window of simulation after stimulation.
Data are reported as mean±s.e.m. Significance of correlations was determined by corrcoef function in Matlab with 0.05 as significance cut-off.
Long-range projections synchronized two cortical networks. (A) Traces of two PYs with LRP conductance of 0 and 0.06. With non-zero LRPs, UP states in PYs synchronize. (B) Power spectrum of PY network activity (red: G(LRP) = 0; blue: G(LRP) = 0.06). LRPs had little effect on overall structure of spectrum but modestly increased peak power.
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Comparison of homologous and non-homologous LRPs (zero delay). (A) Activity snapshots. (B) Phase-plane representation. (C) Correlations between the two PY networks. (D–F) Same representation for non-homologous LRPs.
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Wider variance of delays stabilized networks. Top: Narrow distribution (mean ±20%). Bottom: Wide distribution (mean ±100%). (A) Frequency of state transitions. (B) State distribution of networks. Wider delays result in fewer transitions and a reduced occurrence of non-RF behavior.
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Mechanisms of state transitions. (A) IN activity plots; dashed lines represent example UP states in Network 1. (B) Top and middle: Time snapshots of PY activity in Network 1 and Network 2 for the UP states indicated in (A). Bottom: Synaptic depression variable
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Behavior during tACS in unconnected networks. Distribution of behavior for two networks during tACS with no LRPs (P(local) = 1). Left: Before tACS. Middle: During tACS. Right: After tACS. Spiral waves can still be initiated by tACS even without LRPs, but they are not seen before tACS.
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Percentage of time both networks were in the same state before, during, and after tACS. Left: tACS equalized the time spent in RF across delays. Middle: tACS also increased the likelihood that both networks were in a SP state. Right: tACS biased networks towards simultaneously being in SW (only seen during and after tACS). Values are normalized by the percentage of time spent in each state overall.
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tACS abolished antiphase synchronization. (A) Example of slow antiphase coupling. Upper left: Cross-correlograms between networks before, during, and after tACS. Lower left: Network 2 displayed a state transition before entraining with Network 1 during stimulation. Right: Increased power at 3 Hz in both networks due to tACS. (B) Example of fast antiphase coupling suppressed by tACS. Same plots as in (A).
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Synaptic depression influenced tACS behavior. Left: Depression coefficient at onset of tACS, grouped by behavior during tACS. Lower values indicate more synaptic depression; higher values indicate less synaptic depression. Networks entering SW during tACS had more strongly depressed networks than networks entering RF or SP. Right: Standard deviation of the depression coefficient normalized by the mean. Lower variance of depression correlates with stronger entrainment to tACS.
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Persistence of spiral waves. (A) The three simulations that had constant SW behavior during original simulations, extended for another 7 seconds. One simulation (top) remained in SW except for a brief switch to SP, while the other two simulations (middle and bottom) stayed in SW for the entire time. (B) Persistence of SW in networks that ended with SW after tACS. Simulations ran for another 7 seconds. X-axis indicates the number of seconds SW persisted in the extended period. Many networks leave SW, but 22 networks (28.95%) remain in SW for the entire extended period. (C) Effects of parameters on SW persistence. Lower connectivity (left) correlates with longer persistence of SW. Conductance (middle) and delays (right) have no effect.
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Antiphase induction of high-frequency behavior post-tACS. (A) PY activity during antiphase tACS. During stimulation, the network switches from in-phase ∼3 Hz firing to antiphase firing at 8.6 Hz, persisting upon removal of tACS. (B) Spectrogram shows change from 3 Hz firing to rapid high-frequency firing in both networks.
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Example of rapid fire state. PY activity for two networks is shown in color, indicating the instantaneous firing rate. Most of each network became excited rapidly. Parameters: Delay = 0 msec, P(local) = 0.99, G(LRP) = 0.06.
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Example of slow propagating state. PY activity for two networks in SP for entire simulation. PY activity spread through the network by moving to proximal areas. Parameters: Delay = 50 msec, P(local) = 0.95, G(LRP) = 0.03.
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Example of spiral wave state. Network 1 was in SW for entire simulation while Network 2 was in RF for whole simulation. Spiral wave began with central rotor from which PY activity propagated, forming spiral pattern. Parameters: Delay = 10 msec, P(local) = 0.99, G(LRP) = 0.015.
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tACS entrained networks. Simulation presented in
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tACS disrupted networks. Simulation presented in
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Multistability as a function of propagation delays (
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Distributed delays reduced state transitions (
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Correlation (R2) of dynamics between the two interconnected networks (
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Modulation of state dynamics by simulated tACS as a function of propagation delays (
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Effects of simulation parameters on tACS behavior.
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Synaptic depression contributes to state dynamics. An analysis of the effect of synaptic depression within a network on its response to input from the network connected to it.
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The authors thank Mohsin Ali for code development and helpful input on the manuscript and the Scientific Computing Group at UNC, in particular Dr. Mark Reed, for support with high-performance computing infrastructure.