Abstract
We present a mathematical analysis of networks with integrate-and-fire (IF) neurons with conductance based synapses. Taking into account the realistic fact that the spike time is only known within some finite precision, we propose a model where spikes are effective at times multiple of a characteristic time scale δ, where δ can be arbitrary small (in particular, well beyond the numerical precision). We make a complete mathematical characterization of the model-dynamics and obtain the following results. The asymptotic dynamics is composed by finitely many stable periodic orbits, whose number and period can be arbitrary large and can diverge in a region of the synaptic weights space, traditionally called the “edge of chaos”, a notion mathematically well defined in the present paper. Furthermore, except at the edge of chaos, there is a one-to-one correspondence between the membrane potential trajectories and the raster plot. This shows that the neural code is entirely “in the spikes” in this case. As a key tool, we introduce an order parameter, easy to compute numerically, and closely related to a natural notion of entropy, providing a relevant characterization of the computational capabilities of the network. This allows us to compare the computational capabilities of leaky and IF models and conductance based models. The present study considers networks with constant input, and without time-dependent plasticity, but the framework has been designed for both extensions.
Highlights
Nicolas Brunel, CNRS, FranceNeuronal networks have the capacity to treat incoming information, performing complex computational tasks, including sensory-motor tasks.It is a crucial challenge to understand how this information is encoded and transformed
In the present paper we extend these results, and give a mathematical treatment of the dynamics of spikes generated in synaptic coupled
Mathematical analysis of its asymptotic dynamics has been done in Cessac (2008) and we extend these results to the more delicate case of conductance based IF models in the present paper. [Note that having constant conductances leads to a dynamics which is independent of the past firing times
Summary
Neuronal networks have the capacity to treat incoming information, performing complex computational tasks (see Rieke et al, 1996 for a deep review), including sensory-motor tasks. A complete classification of the dynamical regimes exhibited by this class of IF models was proposed and a one-to-one correspondence between membrane potential trajectories and raster plots was exhibited (for recent contributions that study periodic orbits in large networks of IF neurons, see Gong and van Leeuwen, 2007; Jahnke et al, 2008) Beyond these mathematical results, this work warns one about some conclusions drawn from numerical simulations and emphasizes the necessity to have, when possible, a rigorous analysis of the dynamics. In Cessac (2008) for the BMS model, that there is a sharp transition from fixed point to complex dynamics, when crossing a critical manifold usually called the “edge of chaos” in the literature While this notion is usually not sharply defined in the Neural Network literature, we shall give a mathematical definition which is tractable numerically.
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