Berlin 2008 – scientific programme

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BP: Fachverband Biologische Physik

BP 3: Neuronal Systems

BP 3.8: Talk

Monday, February 25, 2008, 16:15–16:30, C 243

Chaotic dynamics in the balanced state — •Michael Kreissl¹, Siegrid Löwel², and Fred Wolf¹ — ¹Max Planck Institute for Dynamics and Self-Organization, BCCN in Göttingen, Germany, — ²Friedrich Schiller University in Jena, Germany

We study the dynamics of sparse, neural networks in the balanced state. In our networks N Theta-neurons (phase representation of the Quadratic Integrate&Fire model) [Gutkin1998] are pulse-coupled to other neurons with the probability K/N. Using closed expressions for the time evolution of the individual neurons, we perform numerically exact, event based simulations of the network dynamics. Furthermore, we derive the Jacobian of the mapping between spikes analytically, which is used to calculate the long term Lyapunov spectrum through the evolution of a tangential orthonormal system.

Our simulations show that the Lyapunov spectrum in general contains a considerable fraction of positive Lyapunov exponents, indicating chaotic behavior of the network dynamics. The dimension of the attractor is in general large (approx. N/3). The mean Lyapunov exponent is found to be negative, expressing the networks dynamics to be dissipative. In a random matrix approximation, we find an analytic expression of the mean Lyapunov exponent, which is verified by the numerical simulations. We conclude that the balanced state in networks of neurons with active spike generation exhibits conventional and most-probably extensive chaos. This distinguishes such models from binary networks, exhibiting hyperchaos [Vreeswijk1996], and Leaky Integrate&Fire networks, exhibiting stable chaos [Zillmer2006, Jahnke2007].