A method for decoding the neurophysiological spike-response transform

Abstract

Many physiological responses elicited by neuronal spikes—intracellular calcium transients, synaptic potentials, muscle contractions—are built up of discrete, elementary responses to each spike. However, the spikes occur in trains of arbitrary temporal complexity, and each elementary response not only sums with previous ones, but can itself be modified by the previous history of the activity. A basic goal in system identification is to characterize the spike-response transform in terms of a small number of functions—the elementary response kernel and additional kernels or functions that describe the dependence on previous history—that will predict the response to any arbitrary spike train. Here we do this by developing further and generalizing the “synaptic decoding” approach of Sen et al. (1996). Given the spike times in a train and the observed overall response, we use least-squares minimization to construct the best estimated response and at the same time best estimates of the elementary response kernel and the other functions that characterize the spike-response transform. We avoid the need for any specific initial assumptions about these functions by using techniques of mathematical analysis and linear algebra that allow us to solve simultaneously for all of the numerical function values treated as independent parameters. The functions are such that they may be interpreted mechanistically. We examine the performance of the method as applied to synthetic data. We then use the method to decode real synaptic and muscle contraction transforms.

Introduction

The spike is a basic organizing unit of neural activity. Each discrete spike stimulus elicits a discrete elementary response that propagates through various neurophysiological pathways. The prototypical example is at the synapse, where each presynaptic spike elicits, in turn, a discrete rise in presynaptic calcium concentration, release of neurotransmitter, and changes in postsynaptic conductance, current, and potential (Katz, 1969, Johnston and Wu, 1995, Sabatini and Regehr, 1999). The fundamental difficulty in understanding the stimulus–response relationship in such cases comes from the fact that the spikes occur in trains of arbitrary temporal complexity, and that each elementary response not only sums with previous ones, but can itself be greatly modified by the previous history of the activity—in synaptic physiology, by the well-known processes of synaptic plasticity such as facilitation, depression, and post-tetanic potentiation (Magleby and Zengel, 1975, Magleby and Zengel, 1982, Krausz and Friesen, 1977, Zengel and Magleby, 1982, Sen et al., 1996, Hunter and Milton, 2001, Zucker and Regehr, 2002). Downstream of but implicitly including these processes of synaptic plasticity, when they occur at a neuromuscular junction, is then such a response as muscle contraction. In this paper we will analyze, in addition to synthetic data, experimental data from a “slow” invertebrate muscle where prolonged response summation and a highly nonlinear and plastic neuromuscular transform (Brezina et al., 2000) make for a very complex, irregular response that, at first sight, appears extremely challenging to understand and predict quantitatively. Fig. 1 illustrates some of the factors that are responsible for the complexity of the spike-response transform with synthetic data.

To achieve a predictive understanding of the spike-response transform is our goal here. As indicated in Fig. 1, from the observable data, just the spike train and overall response to it, we wish to extract a set of building-block functions—the elementary response kernel and other functions that describe the dependence of the response on the previous history—that will allow us to predict quantitatively the response not only to that particular spike train, but to any arbitrary spike train. This will constitute a complete spike-level characterization of the spike-response transform.

The problem is one of nonlinear system identification. In neurophysiology, and synaptic physiology in particular, such problems have been approached by several methods (for a comparative overview see Marmarelis, 2004). A classic method is to fit the data with a specific model (e.g., Magleby and Zengel, 1975, Magleby and Zengel, 1982, Zengel and Magleby, 1982). However, the choice of model must typically be guided by the limited dataset itself, so that often the model fails to generalize. In a model-free approach, on the other hand, white-noise stimulation is used to determine the system’s Volterra, Wiener, or other similar kernel expansion (e.g., Marmarelis and Naka, 1973, Krausz and Friesen, 1977, Gamble and DiCaprio, 2003, Song et al., 2009a, Song et al., 2009b; for reviews see Sakai, 1992, Marmarelis, 2004). Although in principle providing a complete, general characterization, the higher order kernels are difficult to compute, visualize, and interpret mechanistically. To combine the strengths and minimize the drawbacks of these two approaches, Sen et al. (1996) introduced, and Hunter and Milton (2001) extended, the method of “synaptic decoding.” This method follows the model-free approach as far as to find the system’s first-order, linear kernel (the elementary response kernel), but then, rather than computing the higher order kernels, combines the first-order kernel with a small number of additional linear kernels and simple functions—thus constituting a model, but a relatively general one—to account for the higher order nonlinearities. Here we adopt this basic strategy. We cannot adopt, however, the simplifications that Sen et al. and Hunter and Milton were able to make by virtue of the fact that their decoding method was geared toward synaptic physiology, where, for example, some function forms were a priori more plausible than others. With the fast and rarely summating synaptic responses, they were furthermore able to obtain the shape of the elementary response kernel and the amplitude to which it was scaled at each spike essentially by inspection (Hunter and Milton, 2001). In many slow neuromuscular systems, in contrast, an isolated single spike produces no contraction at all, and when spikes are sufficiently close together to produce contractions, the contractions summate and fuse, so that the elementary response can never be seen in isolation. (Fig. 1 shows this with synthetic data.) Finally, Sen et al. and Hunter and Milton extracted parameters by a gradient descent search, requiring many iterations and a good initial guess. Here we describe, and apply to synaptic and neuromuscular data, a decoding method that largely avoids such simplifying assumptions and limitations.

Following Sen et al. (1996), we use the term “decoding” as a convenient shorthand for the process of system identification of the spike-response transform from its input and output, the spike train and the response to it. However, the formulation of our method then offers the possibility of a decoding also in the more usual sense, of the spike train from the response in which, through the transform, the spike train has been encoded (see Section 4.3).