Neuroscience/Computational Neurobiology/Stimulus-response systems

One of the most commonly studied systems in neuroscience is the stimulus-response system. In this framework, a system is provided with an input, called a stimulus, and the resulting response of the system is measured. Some typical examples of stimuli are visual scenes, auditory recordings, or electric current (if a neural system is being stimulated directly); typical examples of responses include changes in behavior, membrane potential or firing rate (if a cell [or group of cells] is being recorded from directly). The general goal is to find a function that accurately describes the relation between stimulus and response.

Single-neuron systems
In the single neuron system one attempts to characterize how an electric current input to a single neuron determines the probability that the cell will fire an action potential, or spike. A single input consists of a time-dependent current, and a single output consists of a sequence of spikes at various times. By examining the relation between spike times and the preceding stimuli, one can attempt to identify stimulus features that drive the cell towards action potential generation. The neuron in question can either be real or simulated.

Spike-triggered average
One method for relating input current to spike probability is to characterize the average stimulus over a certain window of duration $$T$$ preceding each spike. The collection of all stimuli that lead to spike generation is called the spike-triggered ensemble, and the mean of the spike-triggered ensemble is called the spike-triggered average. If one considers the spike-triggered ensemble to represent a probability distribution over important stimuli, then the spike-triggered average is an estimate of the first moment of that distribution. Intuitively, the shape of the spike-triggered average supposedly corresponds with the shape of the primary feature for which the neuron is selective.

Having obtained the spike-triggered average, one can then perform a dimensionality reduction on the stimulus space in order to describe every possible stimulus by the amount it overlaps with the spike-triggered average. If an arbitrary stimulus is described by a vector $$\boldsymbol{s} = (I_{t-n}, I_{t-n + 1}, ..., I_{t-1}, I_{t})$$, where $$I_{t-i}$$ is the electric current at the $$i$$ th time step before the spike, then this dimensionality reduction represents a mapping $$F: \boldsymbol{s} \rightarrow s_0$$, where $$s = \boldsymbol{s} \cdot \boldsymbol{s}_0 / f_s$$, i.e., the scalar-valued projection of $$\boldsymbol{s}$$ onto the spike-triggered average $$\boldsymbol{s}_0$$, normalized by the sampling frequency $$f_s$$. We can write the probability of a spike occurring in response to a stimulus, $$P(spike|s_0)$$ using Bayes' law:

$$ P(spike|s_0) = \frac{P(s_0|spike)P(spike)}{P(s_0)}. $$

The right hand side of this equation can be easily calculated from the data, as $$P(s_o|spike)$$ is spike-triggered ensemble projected onto $$\boldsymbol{s}_0$$, $$P(s_0)$$ is the entire stimulus ensemble projected onto $$\boldsymbol{s}_0$$, and $$P(spike)$$ is the average firing rate across all stimuli. Thus, to calculate spike probability for a given stimulus, one first filters the stimulus with the spike-triggered average, and then passes the filtered stimulus through a nonlinearity $$P(spike|s)$$ to obtain the final probability. This is an example of the linear-nonlinear-poisson (LNP) model for action potential generation in a neuron.

Spike-triggered covariance
It is often the case that the spike-triggered average provides an insufficient characterization of the response properties of the neuron. For example, consider a neuron that responds both to large increases and large decreases in current. If a white noise stimulus is presented to the neuron, the spike-triggered ensemble will consist of an approximately even mixture of stimuli with large increases in current and stimuli with large decreases in current. The spike-triggered average, however, will consist of a single current trace in which little change is present (because the increases and decreases cancel each other out in the averaging process), thus failing to capture the relevant stimulus features that drive the neuron. However, if one examines the second moment of the spike-triggered ensemble, called the spike-triggered covariance, one can often recover the characteristics of the features important to the neuron. Spike-triggered covariance analysis is related to principal component analysis (PCA) in that it captures the directions in stimulus space with the highest variance.

The reasoning behind spike-triggered covariance analysis is more intuitive in two dimensions. Thus, consider a sinusoidal electric current stimulus in which the only parameters that vary are the frequency and amplitude of the stimulus. Each spike-triggered stimulus in the spike-triggered ensemble can then be represented by a point in the plane, with one axis corresponding to frequency and the other to amplitude. The spike-triggered ensemble is represented by the collection of these points, each of whose corresponding stimulus caused the neuron to spike. Loosely speaking, spike-triggered covariance analysis compares the shape of the spike-triggered ensemble with the shape of the entire stimulus ensemble (as represented in the plane). Thus, by looking not just at the average spike-triggering stimulus, but at the shape of the entire ensemble, spike-triggered covariance analysis can more accurately capture the dynamical properties of stimuli relevant to the neuron.

Mathematically, spike-triggered covariance analysis finds the directions $$\boldsymbol{s}_1, \boldsymbol{s}_2, ..., \boldsymbol{s}_m$$ in stimulus space corresponding to the directions of highest variance in the spike-triggered ensemble. $$\boldsymbol{s}_1, \boldsymbol{s}_2, ..., \boldsymbol{s}_m$$ are determined by finding the eigenvectors of the difference between the spike-triggered covariance matrix $$C$$ and the prior covariance matrix (that is, the covariance matrix of the entire stimulus ensemble) $$C^{prior}$$. The eigenvector of $$C - C^{prior}$$ with the largest eigenvalue is the direction that explains the highest amount of variance, the eigenvector with the second largest eigenvalue is the direction that explains the second highest amount of variance, and so on. Arbitrarily many directions can be used to describe the stimulus space, but typically there will be a point at which adding more stimulus directions contributes only a negligible increase in the predictive capability of the model.

Determining spike probability using covariance analysis proceeds in a manner quite similar to using the spike-triggered average, except that more dimensions are used in the stimulus description. Thus, one first computes $$s_1 = \boldsymbol{s} \cdot \boldsymbol{s}_1/f_s, s_2 = \boldsymbol{s} \cdot \boldsymbol{s}_2/f_s$$, and so forth. Doing this for all stimuli in the spike-triggered ensemble and in the entire stimulus ensemble result in the distributions $$P(s_1,s_2,...,s_m|spike)$$ and $$P(s_1,s_2,...,s_m)$$, respectively. Combining these together with Bayes' law yields

$$ P(spike|s_1,s_2,...,s_m) = \frac{P(s_1,s_2,...,s_m|spike)P(spike)}{P(s_1,s_2,...,s_m)}. $$

Thus, when using spike-triggered covariance analysis, one takes advantage of more stimulus features than the spike-triggered average, therefore allowing a richer description of the properties of stimuli likely to yield a spike.

Generalized linear model
Whereas the linear filters ($$\boldsymbol{s}_0, \boldsymbol{s}_1, ... $$) for the LNP model of spike-generation are computed in a deterministic way when using spike-triggered averages and spike-triggered covariances, in the generalized linear model (GLM), these filters are determined under a probabilistic framework. Specifically, the filters are chosen via maximum likelihood, that is, the filters are chosen that maximize the probability of the observed dataset. Importantly, in the GLM framework, the likelihood function is log-convex, and so a global maximum can be attained, even in a high-dimensional filter space. The firing rate of a neuron in response to a stimulus under the GLM model is given by

$$ \lambda(t) = exp(\boldsymbol{k} \cdot \boldsymbol{s} + \boldsymbol{h} \cdot \boldsymbol{y} + \sum_i \boldsymbol{I}_i \cdot \boldsymbol{y}_i) $$

where $$\boldsymbol{k}$$ is a stimulus filter, $$\boldsymbol{h}$$ is a filter for the cell's spike history, and $$\{\boldsymbol{I}_i\}$$ are the spike-history filters for other relevant cells. Thus, the GLM provides a natural framework for incorporating not only stimulus, but also spike-history dependence of several interacting cells in a network.