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Reverse-engineering in spiking neural networks parameters: exact deterministic parameters estimation

We consider the deterministic evolution of a time-discretized network with spiking neurons, where synaptic transmission has delays, modeled as a neural network of the generalized integrate-and-fire (gIF) type. The purpose is to study a class of algorithmic methods allowing one to calculate the proper parameters to reproduce exactly a given spike train, generated by an hidden (unknown) neural network. This standard problem is known as NP-hard when delays are to be calculated. We propose here a reformulation, now expressed as a Linear-Programming (LP) problem, thus allowing us to provide an efficient resolution. This allows us to “reverse engineer” a neural network, i.e. to find out, given a set of initial conditions, which parameters (i.e., synaptic weights in this case), allow to simulate the network spike dynamics. More precisely we make explicit the fact that the reverse engineering of a spike train, is a Linear (L) problem if the membrane potentials are observed and a LP problem if only spike times are observed. Numerical robustness is discussed. We also explain how it is the use of a generalized IF neuron model instead of a leaky IF model that allows us to derive this algorithm. Furthermore, we point out how the L or LP adjustment mechanism is local to each unit and has the same structure as an “Hebbian” rule. A step further, this paradigm is easily generalizable to the design of input-output spike train transformations. This means that we have a practical method to “program” a spiking network, i.e. find a set of parameters allowing us to exactly reproduce the network output, given an input. Numerical verifications and illustrations are provided.