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Keller, K.; Rimpler, H.; Schneider, G.; Aye, R. D.; Bauer, Irmgard; Baumann, Thomas W.; Beck, M.; Beil, A.; Blascheck, W.; Brand, N.; Brenneisen, Rudolf; Burger, A.; Carle, Reinhold; Chaurasia, N.; Diettrich, B.; Eilert, U.; Ferstl, W.; Fett, Monika; Franck, G.; Frank, B.; Gehrmann, B.; Gorecki, P.; Gracza, L.; Greiner, Sibylle; Grubert, M. W.; Hampe, S.; Hänsel, Rudolf; Harnischfeger, G.; Henkler, G.; Hensel, Andreas; Heubl, Günther; Hiermann, A.; Hiller, K.; Hoffmann-Bohm, K.; Hohmann, B.; Hölscher, U.; Horz, K. H.; Isaac, O.; Junior, Peter; Jüptner, J.; Jurenitsch, J.; Kampfer, M.; Kartnig, Theodor; Khaliefi, F.; Kletter, C.; Kober, M.; Koch, H.; Koehler, H.; Kopp, Brigitte; Kraus, J.; Kraus, Ludwig; Kreis, Wolfgang; Langhammer, L.; Liersch, R.; Lindequist, Ulrike; Loew, D.; Luckner, Martin; Mathys, K.; Merfort, Irmgard; Miething, H.; Moeck, S.; Mundt, S.; Paper, D.; Pilgrim, H.; Proksch, Peter; Ratka, O.; Rauwald, Hans W.; Reichling, J.; Schennen, C.; Schilz, W.; Schleinitz, H.; Schmidt, Sandra; Schneider, K.; Scholz, Eberhard; Schöpke, T.; Schwarze, H.; Schweins, S.; Schwell, B.; Seitz, R.; Siegers, C.‐P.; Simon, P.; Sonnenborn, U.; Ssymank, V.; Staesche, K.; Stahl‐Biskup, E.; Steuding, W.; Stuppner, Hermann; Teuscher, E.; Thober-Miething, H.; Uehleke, B.; Veit, Markus; Warncke, D.; Wink, Michaël; Wissinger-Gräfenhahn, U.; Zepernick, B.

doi:10.1007/978-3-642-58087-1_3

Cited by 3 publications

(4 citation statements)

References 1,297 publications

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“…is characterized by the recurrent inhibitory mean gain parameterḡ < 0, the gain parameter of the synaptic fluctuations, g 2 = N (J ij ) 2 , and the excitatory external input, h 0 > 0. The system is further simplified by assuming a transfer function of a threshold-power law form [33,34],…”

Section: Single Inhibitory Population With Threshold-power Law Tramentioning

confidence: 99%

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Transition to Chaos in Random Neuronal Networks

Kadmon¹,

Sompolinsky²

2015

Phys. Rev. X

175

384

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Firing patterns in the central nervous system often exhibit strong temporal irregularity and considerable heterogeneity in time-averaged response properties. Previous studies suggested that these properties are the outcome of the intrinsic chaotic dynamics of the neural circuits. Indeed, simplified rate-based neuronal networks with synaptic connections drawn from Gaussian distribution and sigmoidal nonlinearity are known to exhibit chaotic dynamics when the synaptic gain (i.e., connection variance) is sufficiently large. In the limit of an infinitely large network, there is a sharp transition from a fixed point to chaos, as the synaptic gain reaches a critical value. Near the onset, chaotic fluctuations are slow, analogous to the ubiquitous, slow irregular fluctuations observed in the firing rates of many cortical circuits. However, the existence of a transition from a fixed point to chaos in neuronal circuit models with more realistic architectures and firing dynamics has not been established. In this work, we investigate rate-based dynamics of neuronal circuits composed of several subpopulations with randomly diluted connections. Nonzero connections are either positive for excitatory neurons or negative for inhibitory ones, while single neuron output is strictly positive with output rates rising as a power law above threshold, in line with known constraints in many biological systems. Using dynamic mean field theory, we find the phase diagram depicting the regimes of stable fixed-point, unstable-dynamic, and chaotic-rate fluctuations. We focus on the latter and characterize the properties of systems near this transition. We show that dilute excitatoryinhibitory architectures exhibit the same onset to chaos as the single population with Gaussian connectivity. In these architectures, the large mean excitatory and inhibitory inputs dynamically balance each other, amplifying the effect of the residual fluctuations. Importantly, the existence of a transition to chaos and its critical properties depend on the shape of the single-neuron nonlinear input-output transfer function, near firing threshold. In particular, for nonlinear transfer functions with a sharp rise near threshold, the transition to chaos disappears in the limit of a large network; instead, the system exhibits chaotic fluctuations even for small synaptic gain. Finally, we investigate transition to chaos in network models with spiking dynamics. We show that when synaptic time constants are slow relative to the mean inverse firing rates, the network undergoes a transition from fast spiking fluctuations with constant rates to a state where the firing rates exhibit chaotic fluctuations, similar to the transition predicted by rate-based dynamics. Systems with finite synaptic time constants and firing rates exhibit a smooth transition from a regime dominated by stationary firing rates to a regime of slow rate fluctuations. This smooth crossover obeys scaling properties, similar to crossover phenomena in statistical mechanics. The theoretical results are sup...

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Section: Single Inhibitory Population With Threshold-power Law Tramentioning

confidence: 99%

“…whereg = g∆ (ν−1)/2 . The values of m and ∆ are evaluated using Eqs (34), (38), and (36). As one increases g , the fixed point becomes unstable.…”

Section: A Fixed Point and Its Stabilitymentioning

confidence: 99%

Transition to Chaos in Random Neuronal Networks

Kadmon¹,

Sompolinsky²

2015

Phys. Rev. X

175

384

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show abstract

“…65,66 The nature of such transition depends on the distribution of frequencies, 67,68 on the features of the coupling, 69,70,71,72 and on the strength of the noise D. 73,74 In this review our interest will be focussed in the pulse-coupled model although under certain conditions it is possible to convert a model of IFO's into a model of phase-coupled oscillators as has been shown by several authors. 75,76,77 In general, three different levels of synchronization can be observed when considering populations. The strongest implies that all the oscillators have the same phase.…”

Section: Synchronizationmentioning

confidence: 99%

On Self-Organized Criticality and Synchronization in Lattice Models of Coupled Dynamical Systems

Pérez

Corral

Dı́az-Guilera

et al. 1996

Int. J. Mod. Phys. B

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Lattice models of coupled dynamical systems lead to a variety of complex behaviors. Between the individual motion of independent units and the collective behavior of members of a population evolving synchronously, there exist more complicated attractors. In some cases, these states are identified with self-organized critical phenomena. In other situations, with clusterization or phase-locking. The conditions leading to such different behaviors in models of integrate-and-fire oscillators and stick-slip processes are reviewed.

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“…Several investigations have been reported on the property of a pair of the HH neurons [21]- [30]. In the network of two HH oscillators coupled by excitatory couplings without time delay, the unit fires periodically in the synchronized state.…”

Section: Introductionmentioning

confidence: 99%

Spike-Train Responses of a Pair of Hodgkin-Huxley Neurons with Time-Delayed Couplings

Hasegawa

2000

J. Phys. Soc. Jpn.

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Model calculations have been performed on the spike-train response of a pair of Hodgkin-Huxley (HH) neurons coupled by recurrent excitatory-excitatory couplings with time delay. The coupled, excitable HH neurons are assumed to receive the two kinds of spike-train inputs: the transient input consisting of M impulses for the finite duration (M: integer) and the sequential input with the constant interspike interval (ISI). The distribution of the output ISI T o shows a rich of variety depending on the coupling strength and the time delay. The comparison is made between the dependence of the output ISI for the transient inputs and that for the sequential inputs.

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C

Cited by 3 publications

References 1,297 publications

Transition to Chaos in Random Neuronal Networks

Transition to Chaos in Random Neuronal Networks

On Self-Organized Criticality and Synchronization in Lattice Models of Coupled Dynamical Systems

Spike-Train Responses of a Pair of Hodgkin-Huxley Neurons with Time-Delayed Couplings

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