Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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Phase space structure and chaos of pulse-coupled network dynamical systems

The dynamics of neuronal networks is fundamental to the ability of neuronal systems to perform cognitive functions such as sensory processing, working memory, and decision making.  In the cerebral cortex dynamically generated activity patterns are, in general, irregular and spatiotemporally complex.  The prevailing theoretical explanation for this irregular activity is that its origin lies in strong fluctuations in inputs that arise from the dynamical balance of excitation and inhibition - known as the balanced state [1].  It is a long standing theme of dynamic brain theories that such complex activity patterns might serve as a rich encoding and processing space for neural computations.  In particular, if the precise timing of every nerve impulse is used for information encoding, the processing capacity of neuronal circuits might be amazingly rich.  We analyzed the state-space structure of spiking neural networks, namely, sparse random networks of inhibitory leaky integrate-and-fire (LIF) neurons in the balanced state [2].  Although capable of generating complex irregular spike sequences, we found that these networks actually exhibit negative-definite Lyapunov spectra.  The spectra are invariant to the network size, hence this stable dynamics is extensive and preserved in the thermodynamic limit.  We find that various state perturbations are predicted to decay extremely fast.  In particular, in the limit of high connectivity, small perturbations to the membrane potentials of the neurons decay as quickly as in isolated cells.  In addition, single-spike perturbations induce only minute responses in the population firing rates that decay on a millisecond time scale.  Surprisingly, however, single-spike perturbations always lead to exponential state separation, causing complete decorrelation of the micro-states.  This rapid decoherence is also established within only a few milliseconds.  By examining the dynamics for arbitrary perturbation sizes, we explain this behavior by a picture of tangled exponentially separating flux tubes composing the phase space of the network.  Our current work is exploring the range of single neuron dynamics and network structures that can support this exotic phase space organization.

Fig. 1: Visualization of the phase-space structure.  Phase-space cross sections for N = 200 and N = 2000, spanned by two random N-dimensional vectors perpendicular to the trajectory. Flux-tube sections are drawn in one color and separated by solid lines.


[1] van Vreeswijk and Sompolinsky, Science, 274, 1726 (1996)
[2] Monteforte and Wolf, Phys. Rev. X., 2, 041007, (2012)

Contact:  Fred Wolf 

Members working within this Project:

 Fred Wolf 

Former Members:

 Rainer Engelken 
 Maximilian Puelma Touzel 
 Agostina Palmigiano 

Selected Publications:

M. Monteforte, and F. Wolf (2012).
Dynamic Flux Tubes Form Reservoirs of Stability in Neuronal Circuits
Physical Review X 2(041007). download file