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Chaotic Balanced State in a Model of Cortical Circuits
 NEURAL COMPUT
, 1998
"... The nature and origin of the temporal irregularity in the electrical activity of cortical neurons in vivo are still not well understood. We consider the hypothesis that this irregularity is due to a balance of excitatory and inhibitory currents into the cortical cells. We study a network model w ..."
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Cited by 83 (1 self)
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The nature and origin of the temporal irregularity in the electrical activity of cortical neurons in vivo are still not well understood. We consider the hypothesis that this irregularity is due to a balance of excitatory and inhibitory currents into the cortical cells. We study a network model with excitatory and inhibitory populations of simple binary units. The internal feedback is mediated by relatively large synaptic strengths, so that the magnitude of the total excitatory as well as inhibitory feedback is much larger than the neuronal threshold. The connectivity is random and sparse. The mean number of connections per unit is large but small compared to the total number of cells in the network. The network also receives a large, temporally regular input from external sources. An analytical solution of the meanfield theory of this model which is exact in the limit of large network size is presented. This theory reveals a new cooperative stationary state of large networks, which we term a balanced state. In this state, a balance between the excitatory and inhibitory inputs emerges dynamically for a wide range of parameters, resulting in a net input whose temporal fluctuations are of the same order as its mean. The internal synaptic inputs act as a strong negative feedback, which linearizes the population responses to the external drive despite the strong nonlinearity of the individual cells. This feedback also greatly stabilizes 1 the system's state and enables it to track a timedependent input on time scales much shorter than the time constant of a single cell. The spatiotemporal statistics of the balanced state is calculated. It is shown that the autocorrelations decay on a short time scale yielding an approximate Poissonian temporal s...
Chaos and Synchrony in a Model of a Hypercolumn in Visual Cortex
 JOURNAL OF COMPUTATIONAL NEUROSCIENCE 3, 734 (1996)' @ 1996 KLUWER ACADEMIC PUBLISHERS. MANUFACTURED IN THE NETHERLANDS.
, 1996
"... Neurons in cortical slices emit spikes or bursts of spikes regularly in response to a suprathreshold current injection. This behavior is in marked contrast to the behavior of cortical neurons in vivo, whose response to electrical or sensory input displays a strong degree of irregularity. Correlation ..."
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Cited by 41 (6 self)
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Neurons in cortical slices emit spikes or bursts of spikes regularly in response to a suprathreshold current injection. This behavior is in marked contrast to the behavior of cortical neurons in vivo, whose response to electrical or sensory input displays a strong degree of irregularity. Correlation measurements show a significant degree of synchrony in the temporal fluctuations of neuronal activities in cortex. We explore the hypothesis that these phenomena are the result of the synchronized chaos generated by the deterministic dynamics of local cortical networks. A model of a "hypercolumn " in the visual cortex is studied. It consists of two populations of neurons, one inhibitory and one excitatory. The dynamics of the neurons is based on a HodgkinHuxley type model of excitable voltageclamped cells with several cellular and synaptic conductances. A slow potassium current is included in the dynamics of the excitatory population to reproduce the observed adaptation of the spike trains emitted by these neurons. The pattern of connectivity has a spatial structure which is correlated with the internal organization of hypercolumns in orientation columns. Numerical simulations of the model show that in an appropriate parameter range, the network settles in a synchronous chaotic state, characterized by a strong temporal variability ofthe neural activity which is correlated across the hypercolumn. Strong inhibitory feedback is essential for the stabilization of this state. These results show that the cooperative dynamics of large neuronal networks are capable of generating variability and synchrony similar to those observed in cortex. Autocorrelation and crosscorrelation functions of
On Numerical Simulations of IntegrateandFire Neural Networks
, 1998
"... It is shown that very small time steps are required to correctly reproduce the synchronization properties of large networks of integrateandfire neurons when the differential system describing their dynamics is integrated with the standard Euler or second order RungeKutta algorithms. The reason fo ..."
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Cited by 37 (1 self)
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It is shown that very small time steps are required to correctly reproduce the synchronization properties of large networks of integrateandfire neurons when the differential system describing their dynamics is integrated with the standard Euler or second order RungeKutta algorithms. The reason for that behavior is analyzed and a simple improvement of these algorithms is proposed. 1 Introduction Our theoretical understanding of the properties of large neuronal systems relies heavily on simulations of networks consisting of up to several thousands interacting neurons. Such simulations are highly time consuming: for a general network architecture, the CPU time is dominated by the evaluation of the interactions between neurons and scales like N 2 , where N is the size of the network. Moreover, one frequently needs to investigate the system's behavior for many different sets of parameters or to perform a statistical analysis over many initial conditions or samples of the internal no...
The Number of Synaptic Inputs and the Synchrony of Large Sparse Neuronal Networks
, 1999
"... The prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchronize large populations of neurons. We study synchronization in models of large networks of spiking neurons with random sparse connectivity. Synchrony oc ..."
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Cited by 35 (1 self)
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The prevalence of coherent oscillations in various frequency ranges in the central nervous system raises the question of the mechanisms that synchronize large populations of neurons. We study synchronization in models of large networks of spiking neurons with random sparse connectivity. Synchrony occurs only when the average number of synapses, M , that a cell receives is larger than a critical value, M c . Below M c , the system is in an asynchronous state. In the limit of weak coupling, assuming identical neurons, we reduce the model to a system of phase oscillators which are coupled via an effective interaction, \Gamma. In this framework, we develop an approximate theory for sparse networks of identical neurons to estimate M c analytically from the Fourier coefficients of \Gamma. Our approach relies on the assumption that the dynamics of a neuron depend mainly on the number of cells that are presynaptic to it. We apply this theory to compute M c for a model of inhibitory networks of integrateandfire (I&F) neurons as a function of the intrinsic neuronal properties (e.g., the refractory period T r ), the synaptic time constants and the strength of the external stimulus, I ext . The number M c is found to be nonmonotonous with the strength of I ext . For T r = 0, we estimate the minimum value of M c over all the parameters of the model to be 363:8. Above M c , the neurons tend to fire in: 1) smeared one cluster states at high firing rates and 2) smeared two or more cluster states at low firing rates. Refractoriness decreases M c at intermediate and high firing rates. These results are compared against numerical simulations. We show numerically that systems with different sizes, N , behave in the same way provided the connectivity, M , is such a way that 1=M eff = 1=...
Synchronization and Desynchronization in a Network of Locally Coupled WilsonCowan Oscillators
, 1996
"... A network of WilsonCowan oscillators is constructed, and its emergent properties of synchronization and desynchronization are investigated by both computer simulation and formal analysis. The network is a twodimensional matrix, where each oscillator is coupled only to its neighbors. We show analyt ..."
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Cited by 20 (1 self)
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A network of WilsonCowan oscillators is constructed, and its emergent properties of synchronization and desynchronization are investigated by both computer simulation and formal analysis. The network is a twodimensional matrix, where each oscillator is coupled only to its neighbors. We show analytically that a chain of locally coupled oscillators (the piecewise linear approximation to the WilsonCowan oscillator) synchronizes, and present a technique to rapidly entrain finite numbers of oscillators. The coupling strengths change on a fast time scale based on a Hebbian rule. A global separator is introduced which receives input from and sends feedback to each oscillator in the matrix. The global separator is used to desynchronize different oscillator groups. Unlike many other models, the properties of this network emerge from local connections, that preserve spatial relationships among components, and are critical for encoding Gestalt principles of feature grouping. The ability to sy...
Synchronous and Asynchronous Chaos in Coupled Neuromodules
 International Journal of Bifurcation and Chaos
, 1999
"... The parametrized timediscrete dynamics of two recurrently coupled neuromodules is studied analytically and by computer simulations. Conditions for the existence of synchronized dynamics are derived and periodic as well as quasiperiodic and chaotic attractors constrained to a synchronization man ..."
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Cited by 7 (6 self)
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The parametrized timediscrete dynamics of two recurrently coupled neuromodules is studied analytically and by computer simulations. Conditions for the existence of synchronized dynamics are derived and periodic as well as quasiperiodic and chaotic attractors constrained to a synchronization manifold M are observed. Stability properties of the synchronized dynamics is discussed by using Lyapunov exponents parallel and transversal to the synchronization manifold. Simulation results are presented for selected sets of parameters. It is observed that locally stable synchronous dynamics often coexists with asynchronous periodic, quasiperiodic or even chaotic attractors. MPIMIS preprint 24/99, and International Journal of Bifurcation and Chaos, 9, 19571968 (1999). 1 1 Introduction Ever since the feasibility of synchronizing chaotic systems has been established also by Pecora & Carroll [1990], this phenomenon has been investigated in many articles. Part of the work on synch...
Synchronized chaos and other coherent states for two coupled neurons
 Physica D
, 1999
"... Synchronized chaos and other coherent states for two coupled neurons by ..."
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Cited by 7 (3 self)
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Synchronized chaos and other coherent states for two coupled neurons by
Synchronous Chaos in Highdimensional Modular Neural Networks
 Int. J. Bifurcat. Chaos
, 1996
"... The relationship between certain types of highdimensional neural networks and lowdimensional prototypical equations (neuromodules) is investigated. The highdimensional systems consist of nitely many pools containing identical, dissipative and nonlinear singleunits operating in discrete time. ..."
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Cited by 5 (5 self)
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The relationship between certain types of highdimensional neural networks and lowdimensional prototypical equations (neuromodules) is investigated. The highdimensional systems consist of nitely many pools containing identical, dissipative and nonlinear singleunits operating in discrete time. Under the assumption of random connections inside and between pools, the system can be reduced to a set of only a few equations, which  asymptotically in time and system size  describe the behavior of every single unit arbitrarily well. This result can be viewed as synchronization of the single units in each pool. It is stated as a theorem on systems of nonlinear coupled maps, which gives explicit conditions on the single unit dynamics and the nature of the random connections. As an application we compare a 2pool network with the corresponding 2dimensional dynamics. The bifurcation diagrams of both systems become very similar even for moderate system size (N=50) and large disor...
Changing Excitation And Inhibition In Simulated Neural Networks
, 2003
"... The development of synchronous bursting in neuronal ensembles represents an important change in network behavior. To determine the influences on development of such synchronous bursting behavior, we study the dynamics of small networks of sparsely connected excitatory and inhibitory neurons using nu ..."
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Cited by 5 (2 self)
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The development of synchronous bursting in neuronal ensembles represents an important change in network behavior. To determine the influences on development of such synchronous bursting behavior, we study the dynamics of small networks of sparsely connected excitatory and inhibitory neurons using numerical simulations. The synchronized bursting activities in networks evoked by background spikes are investigated. Specifically, patterns of bursting activity are examined when the balance between excitation and inhibition on neuronal inputs is varied and the fraction of inhibitory neurons in the network is changed. For quantitative comparison of bursting activities in networks, measures of the degree of synchrony are used. We demonstrate how changes in the strength of excitation on inputs of neurons can be compensated by change in the strength of inhibition without changing the degree of synchrony in the network. The effect of changing of several network parameters on the network activity is analyzed and discussed. These changes may underlie the transition of network activity from normal to potentially pathologic (e.g. epileptic) states.
A Simple Chaotic Neuron
 Physica D
, 1997
"... The discrete dynamics of a dissipative nonlinear model neuron with selfinteraction is discussed. For units with selfexcitatory connection hysteresis effects, i.e. bistability over certain parameter domains, are observed. Numerical simulations demonstrate that selfinhibitory units with nonzero ..."
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Cited by 4 (1 self)
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The discrete dynamics of a dissipative nonlinear model neuron with selfinteraction is discussed. For units with selfexcitatory connection hysteresis effects, i.e. bistability over certain parameter domains, are observed. Numerical simulations demonstrate that selfinhibitory units with nonzero decay rates exhibit complex dynamics including period doubling routes to chaos. These units may be used as basic elements for networks with higherorder information processing capabilities. appeared in: Physica D, 104, 205  211, 1997. 1 1 Introduction Biological neurons exhibit a large variety of dynamical behaviors even when they are not embedded in a network. This type of dynamics is captured by biologically inspired neuron models like the HodgkinHuxley [1] or the FitzHughNagumo equations [2], [3]. On the other hand, formal neurons used in artificial neural networks like the McCullochPitts neuron or the graded response neurons [4] have only trivial, i.e. convergent dynamics as s...