Copyright, 2008, R.E. Kass, E.N. Brown, and U. Eden REPRODUCTION OR CIRCULATION REQUIRES PERMISSION OF THE AUTHORS Chapter 21 Point Processes Let us return to the spike train data recorded from the supplemental eye field (SEF) shown in Figure 21.1. The raster plot displays a set of sequences of time points at which action potentials occurred, i.e., a set of spike trains. There is considerable irregularity in the spike times, both within and across trials, which may be described in terms of probability models. For sequences of event times, however, we need a richer structure than we have developed so far. The probability models we use in this context are called point processes, reflecting the localization of the events as points in time (or, sometimes, points in space) together with the notion that the probability distributions evolve across time according to a stochastic process. The simplest of point processes are called homogeneous Poisson processes. As we outline in Sec- tions 21.0.3, homogeneous Poisson processes can describe highly irregular sequences of event times that have no discernable temporal structure, and they are easy to work with mathematically. When an experimental stim- ulus or behavior is introduced, however, time-varying characteristics of the process become important. In subsequent subsections we indicate ways that more general processes can retain some of the elegance of Poisson processes while gaining the ability to describe a wide variety of phenomena. 1 2 CHAPTER 21. POINT PROCESSES Spatial Pattern trial number -200 0 200 400 600 -200 0 200 400 600 120 120 80 80 40 40 0 0 firing rate per second -200 0 200 400 600 -200 0 200 400 600 Time (ms) Time (ms) Figure 21.1: Raster plot (TOP) and PSTH (BOTTOM) for an SEF neuron under both the external-cue or \spatial" condition (LEFT) and the complex cue or \pattern" condition (RIGHT). Each line in each raster plot contains data from a single trial, that is, a single instance in which the condition was applied. There are 15 trials for each condition. The tick marks represent spike times, i.e., times at which the neuron fired. Spike trains are fundamental to information processing performed by the brain, and point processes form the foundation for distinguishing signal from noise in spike trains. In the remainder of this introduction we discuss several examples of spike-train data. Before we begin, we would like to issue an important warning: point pro- cesses are not the same as continuous-valued stochastic processes, which can take on a continuum of possible values at each point in time. A point process specifies a random sequence of points or, equivalently, a binary indication at every time point as to whether or not an event occurs at that time. Many standard signal-processing techniques are designed primarily for continuous- valued data. Because event-time data are different, alternative methods of analysis are often preferable. 3 Example: Retinal neuron under constant light and environmen- tal conditions Neurons in the retina typically respond to patterns of light displayed over small sections of the visual field. When retinal neurons are grown in culture and held under constant light and environmental condi- tions, however, they will still spontaneously fire action potentials. In a fully functioning retina, this spontaneous activity is sometimes described as back- ground firing activity, which is modulated as a function of visual stimuli. Fig- ure 21.2 shows the spiking activity of one such neuron firing spontaneously over a period of 30 seconds. Even though this neuron is not responding to any explicit stimuli, we can still see structure in its firing activity. Although most of the ISIs are shorter than 20 msec, some are much longer: there is a small second mode in the histogram around 60-120 milliseconds. This sug- gests that the neuron may experience two distinct states, one in which there are bursts of spikes (with short ISIs) and another, more quiescent state (with longer ISIs). From Figure 21.2 we may also get an impression that there may be bursts of activity, with multiple spikes arriving in quick succession of one another. 2 Example: Spiking activity of a primary motor cortical neuron The spiking activity of neurons in primate motor cortex has been shown to relate to intended motor outputs, such as limb reaching movements. Experi- ments where a monkey performs a two-dimensional reach have shown velocity dependent cosine tuning, whereby a motor cortical neuron fires most when the hand moves in a single preferred direction and the intensity drops off as a cosine function of the difference between the intended movement and that preferred direction, and additionally increases with increasing move- ment speed. Figure 3 shows an example of the spiking activity of a neuron in primate motor cortex as a function of hand movement direction during a center-out reaching task. The neuron fires most intensely when the hand moves in a direction about 170 degrees from east. These firing patterns have also been shown to vary as a function of movement speed (Moran & Schwartz, (1999)). 2 Example: Hippocampal place cell Neurons in rodent hippocampus have spatially specific firing properties, whereby the spiking intensity is high- est when the animal is at a specific location in an environment, and falls off as the animal moves further away from that point. Such receptive fields are called place fields, and neurons that have such firing properties are called 4 CHAPTER 21. POINT PROCESSES Figure 21.2: Spontaneous spiking activity of a goldfish retinal neuron in cul- ture under constant light and environmental conditions over 30 seconds. (A) Retinal ganglion cell (taken from web, may be copyrighted) (B) Histogram of interspike intervals and (C) spike train, from a retinal ganglion cell under constant conditions. Figure 21.3: Cosine tuning in primate motor cortex. (A) Spike rasters for a center out task with eight principal directions. (B) Spike count as a function of direction shows a sinusoidal trend. to be re-done 5 Figure 21.4: Movement trajectory (blue) and hippocampal spiking activity (red) of a rat during a free-foraging task in a circular environment. place cells. Figure 4 shows an example of the spiking activity of one such place cell, as a rat executes a free-foraging task in a circular environment. The rat's path through this environment is shown in blue, and the location of the animal at spike times is overlain in red. It is clear that the firing in- tensity is highest slightly to the southwest of the center of the environment, and decreases when the rat moves away from this point. 2 Point processes also arise in imaging. For instance, in PET imaging, a radioisotope that has been incorporated into a metabolically active molecule is introduced into the subject's bloodstream. These molecules become con- centrated in specific tissues and the radioisotopes decay, emitting positrons. These emissions represent a spatiotemporal point process because they are lo- calized occurrences both spatially, throughout the tissue, and in time. After being emitted, the positrons interact with nearby electrons, producing a pair of photons that shoot out in opposite directions and are detected by a circu- lar ring of photosensors. The arrival of photons at each sensor represents a temporal point process and, by characterizing the temporal interactions be- tween arrivals at multiple sensors, it is possible to infer the original location of the positron emission. By observing and inferring the locations of many such occurrences, it is possible to construct an image of specific metabolically active tissues. Point processes have been applied to many physical phenomena outside of neuroscience. For example, temporal point processes have been used to char- acterize the timing and regularity of heart beats (Barbieri and Brown, 2005); 6 CHAPTER 21. POINT PROCESSES to describe geyser eruptions (Azzalini and Bowman, 1990); and to character- ize and predict the locations and times of major earthquakes (Ogata, 1988). 21.0.1 A point process may be specified in terms of event times, inter-event intervals, or event counts. If s1; s2; : : : ; sn are times at which events occur within some time interval we may take xi = si − si−1, i.e., xi is the elapsed time between si−1 and si, and define x1 = s1. This gives the inter-event waiting times xi from the event times and we could reverse the arithmetic to find the event times from a j set of inter-event waiting times x1; : : : ; xn using sj = i=1 xi. In discussing point processes, both of these representations are useful.P In the context of spike trains, s1; s2; : : : ; sn are the spike times, while x1; : : : ; xn are the inter- spike intervals (ISIs). Nearly all of our discussion of event-time sequences will involve modeling of spike train behavior. To represent the variability among the event times we let X1; X2; : : : be a sequence of positive random variables. Then the sequence of random vari- j ables S1; S2; : : : defined by Sj = i=1 Xi is a point process on (0; 1). In fitting point processes to data wePinstead consider finite intervals of time over which the process is observed, and these are usually taken to have the form (0; T ], but for many theoretical purposes it is more convenient to assume the point process ranges across (0; 1). Another useful way to describe a set of event times is in terms of the counts of events observed over time intervals. The event count in a particular time interval may be considered a random variable. For theoretical purposes it is helpful to introduce a function N(t) that counts the total number of events that have occurred up to and including time t.