Linear-nonlinear-Poisson cascade model

The Linear-nonlinear-Poisson (LNP) cascade model is a simplified functional model of neural spike responses. It has been successfully used to describe the response characteristics of neurons in early sensory pathways, especially the visual system. LNP is the implicit model when using reverse correlation or the spike-triggered average to characterize neural responses with white-noise stimuli.

The Linear-Nonlinear-Poisson Cascade Model

There are three stages of the LNP cascade model. The first stage consists of a linear filter, or linear receptive field, which describes how the neuron integrates the stimulus intensity over space and time. The output of this filter is then passed through a nonlinear function, which gives the neuron's instantaneous spike rate as its output. Finally, the instantaneous spike rate is used to generate spikes according to an inhomogeneous Poisson process.

The linear filtering stage performs dimensionality reduction, reducing the high-dimensional spatio-temporal stimulus space to a low-dimensional feature space, within which the neuron computes its response. The nonlinearity then converts the filter output to a spike rate, which must be non-negative, and accounts to nonlinear phenomena such as spike threshold (or rectification) and response saturation. A Poisson spike generator then converts the continuous, scalar-valued spike rate to a series of spike times. Under this model, where the probability of a spike depends only on the instantaneous spike rate.

Mathematical Formulation

single-filter LNP

Let $\mathbf {x}$ denote the spatio-temporal stimulus vector at a particular instant, and $\mathbf {k}$ denote a linear filter (the neuron's linear receptive field), which is a vector with the same number of elements. Let $f$ denote the nonlinearity, a scalar function with non-negative output. Then the LNP model specifies that, in the limit of small time bins,

$P({\textrm {spike}})\propto f(\mathbf {k} \cdot \mathbf {x} )$,

where the constant of proportionality is the size of the bin. This can be stated more precisely as

$P(y{\textrm {~spikes,~in~a~bin~of~size~}}\Delta t)=\Delta tf(\mathbf {k} \cdot \mathbf {x} )e^{-\Delta f(\mathbf {k} \cdot \mathbf {x} )}$

See also

• Spike-triggered average
• Spike-triggered covariance