On Dynamics of Integrate-and-Fire Neural Networks with Conductance Based Synapses

We present a mathematical analysis of a networks with Integrate-and-Fire neurons and adaptive conductances. Taking into account the realistic fact that the spike time is only known within some \textit{finite} precision, we propose a model where spikes are effective at times multiple of a characteristic time scale $\delta$, where $\delta$ can be \textit{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 Integrate-and-Fire 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.Comment: 36 pages, 9 figure

Simulation of networks of spiking neurons: A review of tools and strategies

We review different aspects of the simulation of spiking neural networks. We start by reviewing the different types of simulation strategies and algorithms that are currently implemented. We next review the precision of those simulation strategies, in particular in cases where plasticity depends on the exact timing of the spikes. We overview different simulators and simulation environments presently available (restricted to those freely available, open source and documented). For each simulation tool, its advantages and pitfalls are reviewed, with an aim to allow the reader to identify which simulator is appropriate for a given task. Finally, we provide a series of benchmark simulations of different types of networks of spiking neurons, including Hodgkin-Huxley type, integrate-and-fire models, interacting with current-based or conductance-based synapses, using clock-driven or event-driven integration strategies. The same set of models are implemented on the different simulators, and the codes are made available. The ultimate goal of this review is to provide a resource to facilitate identifying the appropriate integration strategy and simulation tool to use for a given modeling problem related to spiking neural networks.Comment: 49 pages, 24 figures, 1 table; review article, Journal of Computational Neuroscience, in press (2007

Dynamics and spike trains statistics in conductance-based Integrate-and-Fire neural networks with chemical and electric synapses

We investigate the effect of electric synapses (gap junctions) on collective neuronal dynamics and spike statistics in a conductance-based Integrate-and-Fire neural network, driven by a Brownian noise, where conductances depend upon spike history. We compute explicitly the time evolution operator and show that, given the spike-history of the network and the membrane potentials at a given time, the further dynamical evolution can be written in a closed form. We show that spike train statistics is described by a Gibbs distribution whose potential can be approximated with an explicit formula, when the noise is weak. This potential form encompasses existing models for spike trains statistics analysis such as maximum entropy models or Generalized Linear Models (GLM). We also discuss the different types of correlations: those induced by a shared stimulus and those induced by neurons interactions.Comment: 42 pages, 1 figure, submitte

Hardware design of LIF with Latency neuron model with memristive STDP synapses

In this paper, the hardware implementation of a neuromorphic system is presented. This system is composed of a Leaky Integrate-and-Fire with Latency (LIFL) neuron and a Spike-Timing Dependent Plasticity (STDP) synapse. LIFL neuron model allows to encode more information than the common Integrate-and-Fire models, typically considered for neuromorphic implementations. In our system LIFL neuron is implemented using CMOS circuits while memristor is used for the implementation of the STDP synapse. A description of the entire circuit is provided. Finally, the capabilities of the proposed architecture have been evaluated by simulating a motif composed of three neurons and two synapses. The simulation results confirm the validity of the proposed system and its suitability for the design of more complex spiking neural network

Homogeneous Spiking Neuromorphic System for Real-World Pattern Recognition

A neuromorphic chip that combines CMOS analog spiking neurons and memristive synapses offers a promising solution to brain-inspired computing, as it can provide massive neural network parallelism and density. Previous hybrid analog CMOS-memristor approaches required extensive CMOS circuitry for training, and thus eliminated most of the density advantages gained by the adoption of memristor synapses. Further, they used different waveforms for pre and post-synaptic spikes that added undesirable circuit overhead. Here we describe a hardware architecture that can feature a large number of memristor synapses to learn real-world patterns. We present a versatile CMOS neuron that combines integrate-and-fire behavior, drives passive memristors and implements competitive learning in a compact circuit module, and enables in-situ plasticity in the memristor synapses. We demonstrate handwritten-digits recognition using the proposed architecture using transistor-level circuit simulations. As the described neuromorphic architecture is homogeneous, it realizes a fundamental building block for large-scale energy-efficient brain-inspired silicon chips that could lead to next-generation cognitive computing.Comment: This is a preprint of an article accepted for publication in IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol 5, no. 2, June 201

Spike train statistics and Gibbs distributions

This paper is based on a lecture given in the LACONEU summer school, Valparaiso, January 2012. We introduce Gibbs distribution in a general setting, including non stationary dynamics, and present then three examples of such Gibbs distributions, in the context of neural networks spike train statistics: (i) Maximum entropy model with spatio-temporal constraints; (ii) Generalized Linear Models; (iii) Conductance based Inte- grate and Fire model with chemical synapses and gap junctions.Comment: 23 pages, submitte

Frequency control in synchronized networks of inhibitory neurons

We analyze the control of frequency for a synchronized inhibitory neuronal network. The analysis is done for a reduced membrane model with a biophysically-based synaptic influence. We argue that such a reduced model can quantitatively capture the frequency behavior of a larger class of neuronal models. We show that in different parameter regimes, the network frequency depends in different ways on the intrinsic and synaptic time constants. Only in one portion of the parameter space, called `phasic', is the network period proportional to the synaptic decay time. These results are discussed in connection with previous work of the authors, which showed that for mildly heterogeneous networks, the synchrony breaks down, but coherence is preserved much more for systems in the phasic regime than in the other regimes. These results imply that for mildly heterogeneous networks, the existence of a coherent rhythm implies a linear dependence of the network period on synaptic decay time, and a much weaker dependence on the drive to the cells. We give experimental evidence for this conclusion.Comment: 18 pages, 3 figures, Kluwer.sty. J. Comp. Neurosci. (in press). Originally submitted to the neuro-sys archive which was never publicly announced (was 9803001