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 stimulus intensity over space and time. The output of this filter then passes through a nonlinear function, which gives the neuron's instantaneous spike rate as its output. Finally, the 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 converts the filter output to a (non-negative) spike rate, and accounts for nonlinear phenomena such as spike threshold (or rectification) and response saturation. The Poisson spike generator converts the continuous spike rate to a series of spike times, under the assumption that the probability of a spike depends only on the instantaneous spike rate.

The model offers a useful approximation of neural activity, allowing scientists to derive reliable estimates from a mathematically simple formula.

Mathematical formulation

Single-filter LNP

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

${\displaystyle P(\text{spike})\propto f(\mathbf{k}\cdot \mathbf{x})}$.

For finite-sized time bins, this can be stated precisely as the probability of observing y spikes in a single bin:

${\displaystyle P(y\text{~spikes})=\frac{{\left(\mathrm{\Delta }\lambda \right)}^y}{y!}{e}^{-\mathrm{\Delta }\lambda }}$

where ${\displaystyle \lambda =f(\mathbf{k}\cdot \mathbf{x})}$, and ${\displaystyle \mathrm{\Delta }}$ is the bin size.

Multi-filter LNP

For neurons sensitive to multiple dimensions of the stimulus space, the linear stage of the LNP model can be generalized to a bank of linear filters, and the nonlinearity becomes a function of multiple inputs. Let ${\displaystyle {\mathbf{k}}_{\mathbf{1}},{\mathbf{k}}_{\mathbf{2}},\dots ,{\mathbf{k}}_{\mathbf{n}}}$ denote the set of linear filters that capture a neuron's stimulus dependence. Then the multi-filter LNP model is described by

${\displaystyle P(\text{spike})\propto f({\mathbf{k}}_{\mathbf{1}}\cdot \mathbf{x},{\mathbf{k}}_{\mathbf{2}}\cdot \mathbf{x},\dots ,{\mathbf{k}}_{\mathbf{n}}\cdot \mathbf{x})}$

or

${\displaystyle P(\text{spike})\propto f(K\mathbf{x}),}$

where ${\displaystyle K}$ is a matrix whose columns are the filters ${\displaystyle {\mathbf{k}}_{\mathbf{i}}}$.