FIG. 5: (color online) Output distributions of the two neu- rons with highest (blue diamonds) and lowest (red circles) Kullback-Leibler divergence (3) compared to the mean out- put distribution (dashed green line) and the target exponen- tial output distribution (full black line). For a network with N = 500 neurons and a target output mean firing rate (A) µ = 0 . 28 and (B) µ = 0 . 5 (identical for all neurons). Insets: For the same parameters the mean output distribution of the single self-coupled neuron, see Fig. 1. mittency to chaos as a function of µ and system sizes will be given elsewhere. Conclusions. –It is well known [17], that individual and networks of spiking neurons may show bursting behav- ior. Here we showed, that networks of rate encoding neurons are intermittently bursting for low average fir- ing rates, when polyhomeostatically adapting, entering a fully chaotic regime for larger average activity level. Au- tonomous and polyhomeostatic self-control of dynamical systems may hence lead quite in general to non-trivial dynamical states and novel phenomena. Polyhomeostatic optimization is of relevance in a range of fields. Here we have discussed it in the framework of dynamical system theory. Alternative examples are re- source allocation problems, for which a given limited re- source, like time, needs to be allocated to a set of uses, with the target distribution being the percentages of re- source allocated to the respective uses. These ramifica- tions of polyhomeostatic optimization will be discussed in further studies. The data presented in Fig. 5 shows, that the polyhome- ostatically self-organized state of the network results in firing-rate distributions close to the target distribution. This result is quite remarkable. The polyhomeostatic adaption rules (4) are local, viz every neuron adapts its internal parameters a$_{i}$ and b$_{i}$ independently on its own. Intermittency. – Interestingly, the neural activity pre- sented in the inset of Fig. 4 for µ = 0 . 15 shows inter- mittent or bursting behavior. In the quiet laminar peri- ods the neural activity y ( t ) is close to (but slightly be- low) the target mean of 0 . 15, as one would expect for a homeostatically regulated system and the local Lyapunov exponent is negative. The target distribution of firing rates (2) contains however also a small but finite weight for large values for the output y ( t ), which the system achieves through intermittent bursts of activity. In the laminar/bursting periods the gain a ( t ) is below/above threshold, acting such as a control parameter [18]. Both the intermittent and the chaotic regime are chaotic in terms of the global Lyapunov being positive in both regimes. A qualitative difference can be observed when considering the subspace of activities ( y$_{1}$, . . . , y$_{N}$ ). Evaluating the global Lyapunov exponent in this sub- space, viz the relative time evolution (Δ y$_{1}$, . . . , Δ y$_{N}$ ) of differences in activities, we find this restricted global Lya- punov to be negative in the intermittent and positive in the chaotic regime. Details on the evolution from inter- References [1] G.G. Turrigiano, Trends Neurosci 22 , 221 (1999). [2] G.W. Davis, Annual review of neuroscience 29 , 307 (2006). [3] E. Marder, and J.M. Goaillard, Nature Reviews Neuro- science 7 , 563 (2006). [4] R. Der, U. Steinmetz, and F. Pasemann, in Computa- tional Intelligence for Modelling, Control, and Automa- tion, Concurrent Systems Engineering 55 , 43 (1999). [5] M. Stemmler, C. Koch. Nature Neuroscience 2 , 521 (1999). [6] J. Triesch, in Proceedings of ICANN 2005 , W. Duch et al. (Eds.), LNCS 3696 , 65 (2005). [7] T.M. Cover, J.A. Thomas, Elements of information the- ory , Wiley 2006. [8] H.S. Seung, D.D. Lee, B.Y. Reis, and D.W. Tank, J. Comp. Neuroscience 9 , 171 (2000). [9] J.H. Steil, Neural Networks 20 , 353 (2007). [10] C. Gros, New Journal of Physics 9 , 109 (2007). [11] C. Gros, Cognitive Computation 1 , 77 (2009). [12] M. Rabinovich et al , Phys. Rev. Lett. 87 , 68102 (2001). [13] C. Kirst, and M. Timme, Phys. Rev. E 78 , 065201 (2008). [14] C. Gros, Complex and Adaptive Dynamical Systems, A Primer , Springer (2008). [15] R. Ding, and J. Li, Phys. Lett. A 364 , 396 (2007). [16] H. Sompolinsky, A. Crisanti, and H.H. Sommers, Phys. Rev. Lett. 61 259 (1988). [17] M.A. Arbib, The handbook of brain theory and neural networks , MIT Press (2002). [18] N. Platt, E.A. Spiegel, and C. Tresser, Phys. Rev. Lett. 70 , 279 (1993). 4 0 0.5 1 0 0.01 0.02 0.03 0.04 p(y) 0 1 0 0.02 0.04 0.06 A 0 0.5 1 y 0 0.01 0.02 p(y) 0 1 0 0.02 0.04 0.06 B