Symmetries Constrain Dynamics in a Family of Balanced Neural Networks

Barreiro, Andrea K.; Kutz, J. Nathan; Shlizerman, Eli

doi:10.1186/s13408-017-0052-6

Cited by 4 publications

(13 citation statements)

References 61 publications

(99 reference statements)

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“…In an earlier work, we found an alternative possibility [4]. In examining balanced E-I networks without self-coupling, we persistently observed periodic solutions which could not be explained by random matrix theory.…”

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confidence: 69%

“…The network comprises a total of N neurons, of which n E are excitatory and n I are inhibitory. H is the N × N connectivity matrix; the diagonal entries of H are all 0 to exclude self-interactions of neurons (see [4,Sec. 2.1] for a discussion on why self-coupling of neurons is removed).…”

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confidence: 99%

“…Features of the system such as fixed points and periodic orbits remain when the connectivity matrix is perturbed by a random matrix, i.e. G = H + ǫA for small ǫ, even when the spectrum of G bears no obvious resemblance to the original spectrum of H. We first reported on this phenomenon in [4], where we analyzed all-to-all connected, balanced excitatory-inhibitory networks (n C = 1 and n C I = 1). In this paper, we first flesh out some details about that system: we derive leading order expressions for bifurcation points in the system, for the equilibria near those bifurcation points, and for the Hopf bifurcations that spawn the clustered limit cycles we observed in [4] (section 4).…”

mentioning

confidence: 99%

“…G = H + ǫA for small ǫ, even when the spectrum of G bears no obvious resemblance to the original spectrum of H. We first reported on this phenomenon in [4], where we analyzed all-to-all connected, balanced excitatory-inhibitory networks (n C = 1 and n C I = 1). In this paper, we first flesh out some details about that system: we derive leading order expressions for bifurcation points in the system, for the equilibria near those bifurcation points, and for the Hopf bifurcations that spawn the clustered limit cycles we observed in [4] (section 4). We then extend the analysis to networks in which the excitatory population is split up into clusters (n C > 1, n C I = 1; section 5).…”

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confidence: 99%

“…Since our main interest is locating fixed points and periodic orbits, we can simplify the model using the fact that all excitatory cells within each excitatory cluster must be synchronized at a fixed point or periodic orbit. In the case where there is a single excitatory cluster (n C = 1), if x 1 and x 2 are the activities of two excitatory cells, then a straightforward calculation (see Lemma 3 from [4]) shows that…”

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confidence: 99%

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Bifurcations of a neural network model with symmetry

Parker,

Barreiro

2022

Preprint

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We analyze a family of clustered excitatory-inhibitory neural networks and the underlying bifurcation structures that arise because of permutation symmetries in the network as the global coupling strength g is varied. We primarily consider two network topologies: an all-to-all connected network which excludes selfconnections, and a network in which the excitatory cells are broken into clusters of equal size. Although in both cases the bifurcation structure is determined by symmetries in the system, the behavior of the two systems is qualitatively different. In the all-to-all connected network, the system undergoes Hopf bifurcations leading to periodic orbit solutions; notably, for large g, there is a single, stable periodic orbit solution and no stable fixed points. By contrast, in the clustered network, there are no Hopf bifurcations, and there is a family of stable fixed points for large g.

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confidence: 69%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Bifurcations of a neural network model with symmetry

Parker,

Barreiro

2022

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Symmetry and Noether Theorem for Brain Microcircuits

Bîlteanu

Opriş

2017

The Physics of the Mind and Brain Disorders

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Self-consistent formulations for stochastic nonlinear neuronal dynamics

et al. 2020

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Neuronal networks are interesting physical systems in various respects: they operate outside thermodynamic equilibrium [1], a consequence of directed synaptic connections that prohibit detailed balance [2]; they show relaxational dynamics and hence do not conserve but rather constantly dissipate energy; and they show collective behavior that self-organizes as a result of exposure to structured, correlated inputs and the interaction among their constituents. But their analysis is complicated by three fundamental properties: Neuronal activity is stochastic, the input-output transfer function of single neurons is non-linear, and networks show massive recurrence [3] that gives rise to strong interaction effects. They hence bear similarity with systems that are investigated in the field of (quantum) many particle systems. Here, as well, (quantum) fluctuations need to be taken into account and the challenge is to understand collective phenomena that arise from the non-linear interaction of their constituents. Not surprisingly, similar methods can in principle be used to study these two a priori distinct system classes [4][5][6][7][8].But so far, the techniques employed within theoretical neuroscience just begin to harvest this potential. Here we will take essential steps towards this goal. Concretely, we adapt methods from statistical field theory and functional renormalization group techniques to the study of neuronal dynamics.A central motivation for this work is a coherent presentation of the technical machinery, which is well-developed in other fields of physics [9], to study the statistics and in particular phase transitions in stochastic neuronal systems and to provide a bridge between the stochastic dynamics and effective descriptions with reduced complexity.The large number of synaptic inputs to each neuron in a network allows the application of mean-field theory [10-12] to explain many dynamical phenomena, among them first order phase transitions. The transition from a quiescent to a highly active state in a bistable neuronal network is a prime example of a first order phase transition in neuronal networks [12]. The activation of attractors embedded into the connectivity of a Hopfield network is a second [13]. Combined with linear response theory, network fluctuations can quantitatively be described in binary [14][15][16] and in spiking networks [17][18][19][20][21][22]. Also transitions into oscillatory states by Andronov-Hopf bifurcations are in the realm of linear response theory around a mean-field solution [23][24][25][26].Second order phase transitions in neuronal networks are more challenging because the behavior of the system is dominated by fluctuations on all length scales, so that mean-field theory and its systematic correction by loopwise expansion break down [4]. But understanding these transitions is highly interesting from a neuroscientific point of view, because networks then show large susceptibility to signals. Moreover, signatures of critical states are found ubiquitously in experiments: Par...

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Symmetries Constrain Dynamics in a Family of Balanced Neural Networks

Cited by 4 publications

References 61 publications

Bifurcations of a neural network model with symmetry

Bifurcations of a neural network model with symmetry

Symmetry and Noether Theorem for Brain Microcircuits

Self-consistent formulations for stochastic nonlinear neuronal dynamics

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