Emergent dynamics in a model of visual cortex

A large-scale neuronal network modelling study: Stimulus size modulates gamma oscillations in the primary visual cortex by long-range connections

Stimulus size modulation of neuronal firing activity is a fundamental property of the primary visual cortex. Numerous biological experiments have shown that stimulus size modulation is affected by multiple factors at different spatiotemporal scales, but the exact pathways and mechanisms remain incompletely understood. In this paper, we establish a large-scale neuronal network model of primary visual cortex with layer 2/3 to study how gamma oscillation properties are modulated by stimulus size and especially how long-range connections affect the modulation as realistic neuronal properties and spatial distributions of synaptic connections are considered. It is shown that long-range horizontal synaptic connections are sufficient to produce dimensional modulation of firing rates and gamma oscillations. In particular, with increasing grating stimulus size, the firing rate increases and then decreases, the peak frequency of gamma oscillations decreases and the spectral power increases. These are consistent with biological experimental observations. Furthermore, we explain in detail how the number and spatial distribution of long-range connections affect the size modulation of gamma oscillations by using the analysis of neuronal firing activity and synaptic current fluctuations. Our results provide a mechanism explanation for size modulation of gamma oscillations in the primary visual cortex and reveal the important and unique role played by long-range connections, which contributes to a deeper understanding of the cognitive function of gamma oscillations in visual cortex.

Learning spiking neuronal networks with artificial neural networks: neural oscillations

First-principles-based modelings have been extremely successful in providing crucial insights and predictions for complex biological functions and phenomena. However, they can be hard to build and expensive to simulate for complex living systems. On the other hand, modern data-driven methods thrive at modeling many types of high-dimensional and noisy data. Still, the training and interpretation of these data-driven models remain challenging. Here, we combine the two types of methods to model stochastic neuronal network oscillations. Specifically, we develop a class of artificial neural networks to provide faithful surrogates to the high-dimensional, nonlinear oscillatory dynamics produced by a spiking neuronal network model. Furthermore, when the training data set is enlarged within a range of parameter choices, the artificial neural networks become generalizable to these parameters, covering cases in distinctly different dynamical regimes. In all, our work opens a new avenue for modeling complex neuronal network dynamics with artificial neural networks.

The Use of Reduced Models to Generate Irregular, Broad-Band Signals That Resemble Brain Rhythms

The brain produces rhythms in a variety of frequency bands. Some are likely by-products of neuronal processes; others are thought to be top-down. Produced entirely naturally, these rhythms have clearly recognizable beats, but they are very far from periodic in the sense of mathematics. The signals are broad-band, episodic, wandering in amplitude and frequency; the rhythm comes and goes, degrading and regenerating. Gamma rhythms, in particular, have been studied by many authors in computational neuroscience, using reduced models as well as networks of hundreds to thousands of integrate-and-fire neurons. All of these models captured successfully the oscillatory nature of gamma rhythms, but the irregular character of gamma in reduced models has not been investigated thoroughly. In this article, we tackle the mathematical question of whether signals with the properties of brain rhythms can be generated from low dimensional dynamical systems. We found that while adding white noise to single periodic cycles can to some degree simulate gamma dynamics, such models tend to be limited in their ability to capture the range of behaviors observed. Using an ODE with two variables inspired by the FitzHugh-Nagumo and Leslie-Gower models, with stochastically varying coefficients designed to control independently amplitude, frequency, and degree of degeneracy, we were able to replicate the qualitative characteristics of natural brain rhythms. To demonstrate model versatility, we simulate the power spectral densities of gamma rhythms in various brain states recorded in experiments.

Model Reduction Captures Stochastic Gamma Oscillations on Low-Dimensional Manifolds

Gamma frequency oscillations (25–140 Hz), observed in the neural activities within many brain regions, have long been regarded as a physiological basis underlying many brain functions, such as memory and attention. Among numerous theoretical and computational modeling studies, gamma oscillations have been found in biologically realistic spiking network models of the primary visual cortex. However, due to its high dimensionality and strong non-linearity, it is generally difficult to perform detailed theoretical analysis of the emergent gamma dynamics. Here we propose a suite of Markovian model reduction methods with varying levels of complexity and apply it to spiking network models exhibiting heterogeneous dynamical regimes, ranging from nearly homogeneous firing to strong synchrony in the gamma band. The reduced models not only successfully reproduce gamma oscillations in the full model, but also exhibit the same dynamical features as we vary parameters. Most remarkably, the invariant measure of the coarse-grained Markov process reveals a two-dimensional surface in state space upon which the gamma dynamics mainly resides. Our results suggest that the statistical features of gamma oscillations strongly depend on the subthreshold neuronal distributions. Because of the generality of the Markovian assumptions, our dimensional reduction methods offer a powerful toolbox for theoretical examinations of other complex cortical spatio-temporal behaviors observed in both neurophysiological experiments and numerical simulations.

Malleability of gamma rhythms enhances population-level correlations

An important problem in systems neuroscience is to understand how information is communicated among brain regions, and it has been proposed that communication is mediated by neuronal oscillations, such as rhythms in the gamma band. We sought to investigate this idea by using a network model with two components, a source (sending) and a target (receiving) component, both built to resemble local populations in the cerebral cortex. To measure the effectiveness of communication, we used population-level correlations in spike times between the source and target. We found that after correcting for a response time that is independent of initial conditions, spike-time correlations between the source and target are significant, due in large measure to the alignment of firing events in their gamma rhythms. But, we also found that regular oscillations cannot produce the results observed in our model simulations of cortical neurons. Surprisingly, it is the irregularity of gamma rhythms, the absence of internal clocks, together with the malleability of these rhythms and their tendency to align with external pulses - features that are known to be present in gamma rhythms in the real cortex - that produced the results observed. These findings and the mechanistic explanations we offered are our primary results. Our secondary result is a mathematical relationship between correlations and the sizes of the samples used for their calculation. As improving technology enables recording simultaneously from increasing numbers of neurons, this relationship could be useful for interpreting results from experimental recordings.

Entropy, mutual information, and systematic measures of structured spiking neural networks

The aim of this paper is to investigate various information-theoretic measures, including entropy, mutual information, and some systematic measures that are based on mutual information, for a class of structured spiking neuronal networks. In order to analyze and compute these information-theoretic measures for large networks, we coarse-grained the data by ignoring the order of spikes that fall into the same small time bin. The resultant coarse-grained entropy mainly captures the information contained in the rhythm produced by a local population of the network. We first show that these information theoretical measures are well-defined and computable by proving stochastic stability and the law of large numbers. Then we use three neuronal network examples, from simple to complex, to investigate these information-theoretic measures. Several analytical and computational results about properties of these information-theoretic measures are given.

Neural Circuit Mechanism Underlying the Feeding Controlled by Insula-Central Amygdala Pathway

The Central nucleus of amygdala (CeA) contains distinct populations of neurons that play opposing roles in feeding. The circuit mechanism of how CeA neurons process information sent from their upstream inputs to regulate feeding is still unclear. Here we show that activation of the neural pathway projecting from insular cortex neurons to the CeA suppresses food intake. Surprisingly, we find that the inputs from insular cortex form excitatory connections with similar strength to all types of CeA neurons. To reconcile this puzzling result, and previous findings, we developed a conductance-based dynamical systems model for the CeA neuronal network. Computer simulations showed that both the intrinsic electrophysiological properties of individual CeA neurons and the overall synaptic organization of the CeA circuit play a functionally significant role in shaping CeA neural dynamics. We successfully identified a specific CeA circuit structure that reproduces the desired circuit output consistent with existing experimentally observed feeding behaviors.

•

Activation of the insular cortex→central amygdala (CeA) pathway suppresses feeding

•

Insular cortex neurons send similar excitatory inputs to different types of CeA neurons

•

Model suggests a required circuit with both late firing and regular spiking cells

•

The circuit model can explain current and previous CeA-mediated feeding behaviors

Extreme value theory of evolving phenomena in complex dynamical systems: Firing cascades in a model of a neural network

We extend the scope of the dynamical theory of extreme values to include phenomena that do not happen instantaneously but evolve over a finite, albeit unknown at the onset, time interval. We consider complex dynamical systems composed of many individual subsystems linked by a network of interactions. As a specific example of the general theory, a model of a neural network, previously introduced by other authors to describe the electrical activity of the cerebral cortex, is analyzed in detail. On the basis of this analysis, we propose a novel definition of a neuronal cascade, a physiological phenomenon of primary importance. We derive extreme value laws for the statistics of these cascades, both from the point of view of exceedances (that satisfy critical scaling theory in a certain regime) and of block maxima.

A case study in the functional consequences of scaling the sizes of realistic cortical models

Neuroscience models come in a wide range of scales and specificity, from mean-field rate models to large-scale networks of spiking neurons. There are potential trade-offs between simplicity and realism, versatility and computational speed. This paper is about large-scale cortical network models, and the question we address is one of scalability: would scaling down cell density impact a network’s ability to reproduce cortical dynamics and function? We investigated this problem using a previously constructed realistic model of the monkey visual cortex that is true to size. Reducing cell density gradually up to 50-fold, we studied changes in model behavior. Size reduction without parameter adjustment was catastrophic. Surprisingly, relatively minor compensation in synaptic weights guided by a theoretical algorithm restored mean firing rates and basic function such as orientation selectivity to models 10-20 times smaller than the real cortex. Not all was normal in the reduced model cortices: intracellular dynamics acquired a character different from that of real neurons, and while the ability to relay feedforward inputs remained intact, reduced models showed signs of deficiency in functions that required dynamical interaction among cortical neurons. These findings are not confined to models of the visual cortex, and modelers should be aware of potential issues that accompany size reduction. Broader implications of this study include the importance of homeostatic maintenance of firing rates, and the functional consequences of feedforward versus recurrent dynamics, ideas that may shed light on other species and on systems suffering cell loss.

With the vast numbers of neurons in the cerebral cortex, models in neuroscience are, for practical reasons, often orders of magnitude smaller than the actual network. We examine in this article the scalability of cortical networks. We find that function and dynamics in a network depend on network size. For illustration, we use a previously constructed realistic model of monkey visual cortex. Reducing the number of cells in the model, we find that small changes in synaptic weights can help maintain firing rates. However, model characteristics change fundamentally in the reduced models. Neurons have abnormal intracellular dynamics. The model becomes dominated by feedforward inputs and is less capable of functions requiring neuronal interaction. Modelers need to be aware of the potential issues with reduced cortical network models.

Stochastic neural field model: multiple firing events and correlations

This paper studies a nonlinear dynamical phenomenon called the multiple firing event (MFE) in a spatially heterogeneous stochastic neural field model, which is extended from that in our previous paper (Li et al. in J Math Biol 78:83-115, 2018). MFEs are a partially synchronized spiking barrages that are believed to be responsible for the Gamma oscillation. Rigorous results about the stochastic stability and the law of large numbers are proved, which further imply the well-definedness and computability of many quantities related to MFEs. Then we devote to study spatial and temporal properties of MFEs. Our key finding is that MFEs are spatially correlated but the spatial correlation decays quickly. Detailed mathematical justifications are made based on our qualitative models that aim to demonstrate the mechanism of MFEs.

A coarse-graining framework for spiking neuronal networks: from strongly-coupled conductance-based integrate-and-fire neurons to augmented systems of ODEs

Homogeneously structured, fluctuation-driven networks of spiking neurons can exhibit a wide variety of dynamical behaviors, ranging from homogeneity to synchrony. We extend our partitioned-ensemble average (PEA) formalism proposed in Zhang et al. (Journal of Computational Neuroscience, 37(1), 81-104, 2014a) to systematically coarse grain the heterogeneous dynamics of strongly coupled, conductance-based integrate-and-fire neuronal networks. The population dynamics models derived here successfully capture the so-called multiple-firing events (MFEs), which emerge naturally in fluctuation-driven networks of strongly coupled neurons. Although these MFEs likely play a crucial role in the generation of the neuronal avalanches observed in vitro and in vivo, the mechanisms underlying these MFEs cannot easily be understood using standard population dynamic models. Using our PEA formalism, we systematically generate a sequence of model reductions, going from Master equations, to Fokker-Planck equations, and finally, to an augmented system of ordinary differential equations. Furthermore, we show that these reductions can faithfully describe the heterogeneous dynamic regimes underlying the generation of MFEs in strongly coupled conductance-based integrate-and-fire neuronal networks.

Rhythm and Synchrony in a Cortical Network Model

We studied mechanisms for cortical gamma-band activity in the cerebral cortex and identified neurobiological factors that affect such activity. This was done by analyzing the behavior of a previously developed, data-driven, large-scale network model that simulated many visual functions of monkey V1 cortex (Chariker et al., 2016). Gamma activity was an emergent property of the model. The model's gamma activity, like that of the real cortex, was (1) episodic, (2) variable in frequency and phase, and (3) graded in power with stimulus variables like orientation. The spike firing of the model's neuronal population was only partially synchronous during multiple firing events (MFEs) that occurred at gamma rates. Detailed analysis of the model's MFEs showed that gamma-band activity was multidimensional in its sources. Most spikes were evoked by excitatory inputs. A large fraction of these inputs came from recurrent excitation within the local circuit, but feedforward and feedback excitation also contributed, either through direct pulsing or by raising the overall baseline. Inhibition was responsible for ending MFEs, but disinhibition led directly to only a small minority of the synchronized spikes. As a potential explanation for the wide range of gamma characteristics observed in different parts of cortex, we found that the relative rise times of AMPA and GABA synaptic conductances have a strong effect on the degree of synchrony in gamma.SIGNIFICANCE STATEMENT Canonical computations used throughout the cerebral cortex are performed in primary visual cortex (V1). Providing theoretical mechanisms for these computations will advance understanding of computation throughout cortex. We studied one dynamical feature, gamma-band rhythms, in a large-scale, data-driven, computational model of monkey V1. Our most significant conclusion is that the sources of gamma band activity are multidimensional. A second major finding is that the relative rise times of excitatory and inhibitory synaptic potentials have strong effects on spike synchrony and peak gamma band power. Insight gained from studying our V1 model can shed light on the functions of other cortical regions.

Orientation Selectivity from Very Sparse LGN Inputs in a Comprehensive Model of Macaque V1 Cortex

A new computational model of the primary visual cortex (V1) of the macaque monkey was constructed to reconcile the visual functions of V1 with anatomical data on its LGN input, the extreme sparseness of which presented serious challenges to theoretically sound explanations of cortical function. We demonstrate that, even with such sparse input, it is possible to produce robust orientation selectivity, as well as continuity in the orientation map. We went beyond that to find plausible dynamic regimes of our new model that emulate simultaneously experimental data for a wide range of V1 phenomena, beginning with orientation selectivity but also including diversity in neuronal responses, bimodal distributions of the modulation ratio (the simple/complex classification), and dynamic signatures, such as gamma-band oscillations. Intracortical interactions play a major role in all aspects of the visual functions of the model.

SIGNIFICANCE STATEMENT

We present the first realistic model that has captured the sparseness of magnocellular LGN inputs to the macaque primary visual cortex and successfully derived orientation selectivity from them. Three implications are (1) even in input layers to the visual cortex, the system is less feedforward and more dominated by intracortical signals than previously thought, (2) interactions among cortical neurons in local populations produce dynamics not explained by single neurons, and (3) such dynamics are important for function. Our model also shows that a comprehensive picture is necessary to explain function, because different visual properties are related. This study points to the need for paradigm shifts in neuroscience modeling: greater emphasis on population dynamics and, where possible, a move toward data-driven, comprehensive models.

Impaired neurogenesis of the dentate gyrus is associated with pattern separation deficits: A computational study

The separation of input patterns received from the entorhinal cortex (EC) by the dentate gyrus (DG) is a well-known critical step of information processing in the hippocampus. Although the role of interneurons in separation pattern efficiency of the DG has been theoretically known, the balance of neurogenesis of excitatory neurons and interneurons as well as its potential role in information processing in the DG is not fully understood. In this work, we study separation efficiency of the DG for different rates of neurogenesis of interneurons and excitatory neurons using a novel computational model in which we assume an increase in the synaptic efficacy between excitatory neurons and interneurons and then its decay over time. Information processing in the EC and DG was simulated as information flow in a two layer feed-forward neural network. The neurogenesis rate was modeled as the percentage of new born neurons added to the neuronal population in each time bin. The results show an important role of an optimal neurogenesis rate of interneurons and excitatory neurons in the DG in efficient separation of inputs from the EC in pattern separation tasks. The model predicts that any deviation of the optimal values of neurogenesis rates leads to different decreased levels of the separation deficits of the DG which influences its function to encode memory.

Intrinsic and Network Mechanisms Constrain Neural Synchrony in the Moth Antennal Lobe

Projection-neurons (PNs) within the antennal lobe (AL) of the hawkmoth respond vigorously to odor stimulation, with each vigorous response followed by a ~1 s period of suppression—dubbed the “afterhyperpolarization-phase,” or AHP-phase. Prior evidence indicates that this AHP-phase is important for the processing of odors, but the mechanisms underlying this phase and its function remain unknown. We investigate this issue. Beginning with several physiological experiments, we find that pharmacological manipulation of the AL yields surprising results. Specifically, (a) the application of picrotoxin (PTX) lengthens the AHP-phase and reduces PN activity, whereas (b) the application of Bicuculline-methiodide (BIC) reduces the AHP-phase and increases PN activity. These results are curious, as both PTX and BIC are inhibitory-receptor antagonists. To resolve this conundrum, we speculate that perhaps (a) PTX reduces PN activity through a disinhibitory circuit involving a heterogeneous population of local-neurons, and (b) BIC acts to hamper certain intrinsic currents within the PNs that contribute to the AHP-phase. To probe these hypotheses further we build a computational model of the AL and benchmark our model against our experimental observations. We find that, for parameters which satisfy these benchmarks, our model exhibits a particular kind of synchronous activity: namely, “multiple-firing-events” (MFEs). These MFEs are causally-linked sequences of spikes which emerge stochastically, and turn out to have important dynamical consequences for all the experimentally observed phenomena we used as benchmarks. Taking a step back, we extract a few predictions from our computational model pertaining to the real AL: Some predictions deal with the MFEs we expect to see in the real AL, whereas other predictions involve the runaway synchronization that we expect when BIC-application hampers the AHP-phase. By examining the literature we see support for the former, and we perform some additional experiments to confirm the latter. The confirmation of these predictions validates, at least partially, our initial speculation above. We conclude that the AL is poised in a state of high-gain; ready to respond vigorously to even faint stimuli. After each response the AHP-phase functions to prevent runaway synchronization and to “reset” the AL for another odor-specific response.

Asynchronous Rate Chaos in Spiking Neuronal Circuits

The brain exhibits temporally complex patterns of activity with features similar to those of chaotic systems. Theoretical studies over the last twenty years have described various computational advantages for such regimes in neuronal systems. Nevertheless, it still remains unclear whether chaos requires specific cellular properties or network architectures, or whether it is a generic property of neuronal circuits. We investigate the dynamics of networks of excitatory-inhibitory (EI) spiking neurons with random sparse connectivity operating in the regime of balance of excitation and inhibition. Combining Dynamical Mean-Field Theory with numerical simulations, we show that chaotic, asynchronous firing rate fluctuations emerge generically for sufficiently strong synapses. Two different mechanisms can lead to these chaotic fluctuations. One mechanism relies on slow I-I inhibition which gives rise to slow subthreshold voltage and rate fluctuations. The decorrelation time of these fluctuations is proportional to the time constant of the inhibition. The second mechanism relies on the recurrent E-I-E feedback loop. It requires slow excitation but the inhibition can be fast. In the corresponding dynamical regime all neurons exhibit rate fluctuations on the time scale of the excitation. Another feature of this regime is that the population-averaged firing rate is substantially smaller in the excitatory population than in the inhibitory population. This is not necessarily the case in the I-I mechanism. Finally, we discuss the neurophysiological and computational significance of our results.

A reduction for spiking integrate-and-fire network dynamics ranging from homogeneity to synchrony

In this paper we provide a general methodology for systematically reducing the dynamics of a class of integrate-and-fire networks down to an augmented 4-dimensional system of ordinary-differential-equations. The class of integrate-and-fire networks we focus on are homogeneously-structured, strongly coupled, and fluctuation-driven. Our reduction succeeds where most current firing-rate and population-dynamics models fail because we account for the emergence of 'multiple-firing-events' involving the semi-synchronous firing of many neurons. These multiple-firing-events are largely responsible for the fluctuations generated by the network and, as a result, our reduction faithfully describes many dynamic regimes ranging from homogeneous to synchronous. Our reduction is based on first principles, and provides an analyzable link between the integrate-and-fire network parameters and the relatively low-dimensional dynamics underlying the 4-dimensional augmented ODE.

Emergent spike patterns in neuronal populations

This numerical study documents and analyzes emergent spiking behavior in local neuronal populations. Emphasis is given to a phenomenon we call clustering, by which we refer to a tendency of random groups of neurons large and small to spontaneously coordinate their spiking activity in some fashion. Using a sparsely connected network of integrate-and-fire neurons, we demonstrate that spike clustering occurs ubiquitously in both high firing and low firing regimes. As a practical tool for quantifying such spike patterns, we propose a simple scheme with two parameters, one setting the temporal scale and the other the amount of deviation from the mean to be regarded as significant. Viewing population activity as a sequence of events, meaning relatively brief durations of elevated spiking, separated by inter-event times, we observe that background activity tends to give rise to extremely broad distributions of event sizes and inter-event times, while driving a system imposes a certain regularity on its inter-event times, producing a rhythm consistent with broad-band gamma oscillations. We note also that event sizes and inter-event times decorrelate very quickly. Dynamical analyses supported by numerical evidence are offered.

Dynamical models of cortical circuits

Cortical neurons operate within recurrent neuronal circuits. Dissecting their operation is key to understanding information processing in the cortex and requires transparent and adequate dynamical models of circuit function. Convergent evidence from experimental and theoretical studies indicates that strong feedback inhibition shapes the operating regime of cortical circuits. For circuits operating in inhibition-dominated regimes, mathematical and computational studies over the past several years achieved substantial advances in understanding response modulation and heterogeneity, emergent stimulus selectivity, inter-neuron correlations, and microstate dynamics. The latter indicate a surprisingly strong dependence of the collective circuit dynamics on the features of single neuron action potential generation. New approaches are needed to definitely characterize the cortical operating regime.

Gamma frequency feedback inhibition accounts for key aspects of orientation selectivity in V1

There is now strong evidence that gamma frequency oscillations occur during the engagement of cortical regions. These oscillations involve gamma frequency feedback inhibition. Thus, understanding the properties of this form of inhibition is critical to understanding how excitation and inhibition interact to determine which cells fire and, more generally, how cortex performs computations. In previous work, we argued that gamma frequency inhibition performs a type of winner-take-all computation that obeys simple rules: 1) cells fire if their excitation is within E% of the cell with maximum excitation; 2) E%max is determined by the delay of feedback inhibition and the membrane time constant. This framework was previously applied to the best-studied cortical computation, orientation selectivity of cells in V1. Measurements show that orientation tuning is insensitive to illumination contrast. We showed that this finding can be simply explained by the E%max model. Recently, a new property of orientation selectivity has been discovered: orientation tuning varies with the phase of the gamma oscillation. Here, we show that this too can be simply explained by the E%max model. These successes suggest that simple rules underlie the selection of which cells fire in cortical networks.

Two types of asynchronous activity in networks of excitatory and inhibitory spiking neurons

Asynchronous activity in balanced networks of excitatory and inhibitory neurons is believed to constitute the primary medium for the propagation and transformation of information in the neocortex. Here we show that an unstructured, sparsely connected network of model spiking neurons can display two fundamentally different types of asynchronous activity that imply vastly different computational properties. For weak synaptic couplings, the network at rest is in the well-studied asynchronous state, in which individual neurons fire irregularly at constant rates. In this state, an external input leads to a highly redundant response of different neurons that favors information transmission but hinders more complex computations. For strong couplings, we find that the network at rest displays rich internal dynamics, in which the firing rates of individual neurons fluctuate strongly in time and across neurons. In this regime, the internal dynamics interact with incoming stimuli to provide a substrate for complex information processing and learning.

Distribution of correlated spiking events in a population-based approach for Integrate-and-Fire networks

Randomly connected populations of spiking neurons display a rich variety of dynamics. However, much of the current modeling and theoretical work has focused on two dynamical extremes: on one hand homogeneous dynamics characterized by weak correlations between neurons, and on the other hand total synchrony characterized by large populations firing in unison. In this paper we address the conceptual issue of how to mathematically characterize the partially synchronous "multiple firing events" (MFEs) which manifest in between these two dynamical extremes. We further develop a geometric method for obtaining the distribution of magnitudes of these MFEs by recasting the cascading firing event process as a first-passage time problem, and deriving an analytical approximation of the first passage time density valid for large neuron populations. Thus, we establish a direct link between the voltage distributions of excitatory and inhibitory neurons and the number of neurons firing in an MFE that can be easily integrated into population-based computational methods, thereby bridging the gap between homogeneous firing regimes and total synchrony.