# Sparse Bayesian identification of synaptic connectivity from multi-neuronal spike train data

Junichiro Yoshimoto (Okinawa Institute of Science and Technology), Kenji Doya (Okinawa Institute of Science and Technology)

**Introduction**: Recent
progress in neuronal recording technique allows us to record spike events simultaneously
from a majority of neurons in a local circuit. Here we propose a method for
identifying synaptic connectivity from multi-neuronal spike train data based on
a electrophysiological neuron model and statistical inference.

**Method**: The
method assumes that spikes are generated from stochastic leaky
integrate-and-fire (LIF) neurons [1] connected with multi-exponential postsynaptic
current functions. The model is a special class of linear-nonlinear-Poisson
(LNP) models and the parameters can be fitted to spike train data by maximum
likelihood estimation (MLE) [2]. For a network of *n* neurons with *k*
postsynaptic current functions, the number of parameters is *n*^{2}(*k*+1)+*n*, which makes MLE
prone to over-fitting as the number of recorded neurons increases. To overcome
this issue, we develop a Bayesian estimation algorithm with a hierarchical
prior that promotes sparseness of effective connection parameters. In the
algorithm, the prior distribution of the model parameters is given by a
Gaussian distribution with zero mean vector and diagonal covariance matrix, in
which each non-zero element is regarded as a independent random variable
distributed according to a Gamma distribution. All the random variables in the
hierarchical model are identified as the maximum a posteriori (MAP) estimator,
which is computed by the expectation-maximization (EM) algorithm combined with the
Newton-Raphson method. After the parameter fitting, three types of synaptic
connectivity (excitatory, inhibitory, or none) are finally identified based on
the maximum amplitude of spike response curves reproduced by the model.

**Results**: The
method was applied to two synthetic benchmarks. First, the spike train data
were generated from a network of *n*=6 stochastic
LIF neurons with 4 excitatory and 4 inhibitory connections and the reproducibility
of the model parameters and the spike response curve were tested. The proposed
Bayesian algorithm could achieve smaller variance of estimated model parameters
and reproduced the spike response curves better than the MLE method. In the
second benchmark, spike train data were generated from a network of *n*=15 Hodgkin-Huxley type neurons with 50
excitatory and 25 inhibitory connections and the reproducibility of the types
of synaptic connection was tested. According to the ROC (receiver operating
characteristic) analysis, the method could estimate three types of synaptic
connectivity with a high classification precision (AUC score > 99% in
average for our data set).

**References**

- Shinomoto, S. (2010). Fitting a stochastic spiking model to neuronal current injection data. Neural Networks, 23, 764-769.
- Paninski, L. (2004). Maximum likelihood estimation of cascade point-process neural encoding models. Network-Computation in Neural Systems, 15, 243-262.