- Oral presentation
- Open Access
Chaos in heterogeneous neural networks: I. The critical transition point
© Aljadeff et al; licensee BioMed Central Ltd. 2014
- Published: 21 July 2014
- Network Transition
- Synaptic Weight
- Adult Neurogenesis
- Rate Dynamic
- Computational Capacity
There is accumulating evidence that biological neural networks posses optimal computational capacity when they are at or near a critical point in which the network transitions to a chaotic regime. We derive a formula for the critical point of a general heterogeneous neural network. This formula relates the structure of the network to its critical point. The heterogeneity of the network may describe the spatial structure, a multiplicity of cell types or any selective connectivity rules.
To define the network we divide the N neurons into D groups such that ∑ d = 1...D N d =N. The synaptic weight between neurons i,j (the connectivity matrix element J ij ) is drawn from a centered distribution with standard deviation summarized in a D×D rule matrix N -1/2 G c ( i ) d ( j ) (insets to A, c(i) is the type index of neuron i). The network obeys the standard rate dynamics (d/dt)x i =- x i +∑ j = 1...N J ij tanhx j .
The global behavior of the network changes from a single fixed point to chaos when r=1, r being the radius of the circle that bounds the spectrum of the connectivity matrix (panel A). We derived a formula, in terms of the matrix G and the vector N d , for r that can also be thought of as an effective gain: it is the square root of the maximal eigenvalue of a D×D matrix M whose c,d element is M cd = N -1 N c (G cd ) 2 .
- Sompolinsky H, Crisanti A, Sommers H: Chaos in random neural networks. Phys Rev Lett. 1988, 61: 259-262. 10.1103/PhysRevLett.61.259.View ArticlePubMedGoogle Scholar
- Sussillo D, Abbott LF: Generating Coherent Patterns of Activity from Chaotic Neural Networks. Neuron. 2009, 63: 544-557. 10.1016/j.neuron.2009.07.018.PubMed CentralView ArticlePubMedGoogle Scholar
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