Figure 14

Schematic depicting the causal relationship between the outputs and the recent history of the inputs to a nonlinear dynamical system, in this instance a brain region or voxel. This relationship can be expressed as a Volterra series, which expresses the response or output y(t) as a nonlinear convolution of the inputs u(t), critically without reference to any [hidden] state variables. This series is simply a functional Taylor expansion of y(t) as a function of the inputs over the recent past. is the ith order kernel. Volterra series have been described as a 'power series with memory' and are generally thought of as a high-order or 'nonlinear convolution' of the inputs to provide an output. Volterra kernels are useful in characterizing the effective connectivity or influences that one neuronal system exerts over another because they represent the causal characteristics of the system in question. Neurobiologically they have a simple and compelling interpretation they are synonymous with effective connectivity. It is evident that the first-order kernel embodies the response evoked by a change in input at . In other words it is a time-dependant measure of driving efficacy. Similarly the second order kernel reflects the modulatory influence of the input at on the evoked response at . And so on for higher orders.