How can the central nervous system make accurate decisions about external stimuli at short times on the basis of the noisy responses of nerve cell populations? It has been suggested that spike time latency is the source of fast decisions. Here, we propose a simple and fast readout mechanism, the temporal Winner-Take-All (tWTA), and undertake a study of its accuracy. The tWTA is studied in the framework of a statistical model for the dynamic response of a nerve cell population to an external stimulus. Each cell is characterized by a preferred stimulus, a unique value of the external stimulus for which it responds fastest. The tWTA estimate for the stimulus is the preferred stimulus of the cell that fired the first spike in the entire population. We then pose the questions: How accurate is the tWTA readout? What are the parameters that govern this accuracy? What are the effects of noise correlations and baseline firing? We find that tWTA sensitivity to the stimulus grows algebraically fast with the number of cells in the population, N, in contrast to the logarithmic slow scaling of the conventional rate-WTA sensitivity with N. Noise correlations in first-spike times of different cells can limit the accuracy of the tWTA readout, even in the limit of large N, similar to the effect that has been observed in population coding theory. We show that baseline firing also has a detrimental effect on tWTA accuracy. We suggest a generalization of the tWTA, the n-tWTA, which estimates the stimulus by the identity of the group of cells firing the first n spikes and show how this simple generalization can overcome the detrimental effect of baseline firing. Thus, the tWTA can provide fast and accurate responses discriminating between a small number of alternatives. High accuracy in estimation of a continuous stimulus can be obtained using the n-tWTA.
In recent years, there has been growing interest in coding information about external stimuli by the fine temporal structure of the neural dynamic response [1]–[18]. Several studies have shown that response latency is modulated by external stimuli [1]–[4]. Many cells in the middle temporal (MT) cortex code for the direction of motion of visual stimuli, and can be characterized by a preferred direction of the stimulus, to which they respond maximally, see e. g. , [19], [20]. Osborne et al. [1] reported that the MT cells respond with the shortest delay when the stimulus is in their preferred direction and that the response delay increases as the stimulus direction diverges from the preferred direction of the cell. In the auditory system of the ferret, Nelken et al. [2] showed response-latency tuning in primary auditory cortex cells to the direction of a virtual sound source. In a recent work Gollisch and Meister [18] showed that relative first-spike times of retinal ganglion cells carry considerable information about the external stimulus, but they did not suggest a concrete readout mechanism. Here we study the accuracy of a simple readout mechanism, the temporal-Winner-Take-All (tWTA), which extracts information from response latency. The tWTA estimates the stimulus by the identity of the cell that fired the first spike in a population of cells, in contrast to the conventional rate-Winner-Take-All (WTA), which estimates the stimulus by the identity of the cell that fired the most spikes. For example, the tWTA estimate for the direction of motion of a visual stimulus from the responses of a population of MT cells would be the preferred direction of the cell that fired the first spike in the entire population. Considerable theoretical effort has been devoted to the study of the accuracy of population code readout mechanisms, such as the population-vector, optimal-linear and ideal observer readouts. Of particular interest in the investigation of these readouts was the dependence of the readout accuracy on the population size and the effects of noise correlations in the neuronal responses. In this work, we quantify tWTA accuracy. To this end, we address three specific questions. One, what are the essential features of the neuronal dynamic response to the stimulus to which the tWTA is sensitive? Two, how does the tWTA accuracy depend on the population size? Three, what are the effects of noise correlations and baseline firing on tWTA accuracy? These questions are addressed in the framework of a statistical model for the dynamic response of MT cells to a moving visual stimulus. In the first part of the results section we investigate tWTA accuracy in a two-column competition model, and in the second part we study tWTA accuracy in the framework of a hypercolumn model. Both parts start by defining the statistical model of the neuronal dynamic response and then follow with an investigation of tWTA accuracy in the absence of noise correlations and baseline firing. In the final stage of each part, correlations and baseline firing are introduced and their effect on tWTA accuracy is investigated.
Substituting equation (14) into equation (4), we obtain the probability of correct discrimination as sum of two terms: (39) (40) (41) The integral, equation (40), can be evaluated exactly, yielding the contribution of (42) where we have used as shorthand for. The contribution of to is positive; hence, the tWTA error rate, in this case, will decay to zero exponentially with the population size. For the calculation of, equation (41), we change variables to, yielding: (43) where for a small positive, . Assuming, the leading term in, for small, is given by: (44) Using Watson' s Lemma to evaluate to leading order in, we obtain (45) where (46) and (47) For a given stimulus, the probability density, , that the first spike of population 1 occurred at time is (48) where. The probability density, , that population 2 did not fire before time is given by: (49) Assuming the tWTA decides the stimulus is if the first spike comes from population 1, the probability of correct response is: (50) In the limit of large, , we obtain, and due to the delay, we thus obtain (51) In the specific example of Figure 3, the following model was used. The uncorrelated part of the first spike times was generated by an IHPP with a delayed step function PSTH, yielding the first-spike-time probability density: with and. For the correlated part, we used an IHPP model with PSTH, yielding: . From equation (48), the probability density, , that the first spike in column 1 occurred at time, is given by: (52) The probability density, , that no cell in column 2 had fired until time, equation (49), is equal to the probability that the first spike of the cells in column 2 occurred at any time: (53) Note that for, . The probability of correct classification is given by: (54) Substituting equations (52) and (53) into equation (54) and integrating one obtains the result of equation (20). We now turn to calculate tWTA asymptotic accuracy in the presence of correlations, see section ‘Effect of correlations on the tWTA estimation accuracy’. In the limit of large, the estimation error will be dominated by the correlated noise. We can, therefore, neglect the fluctuations of the uncorrelated part, (see equation (34) ), replacing its distribution with a delta function at the first time the PSTH of cell is larger than zero: (55) In this case, for a specific realization of, and we can write the first spike time of cell as (56) Without loss of generality, we will assume. The tWTA estimate for the stimulus, is obtained by taking the derivative of equation (56) with respect to and equating to zero at (57) For small errors, we can approximate: (58) and obtain: (59) which is equivalent to the result of equation (35).
Considerable experimental as well as theoretical effort has been devoted to the investigation of the neural code. The traditional approach has been to study the information content of the total neural spike count during a long period of time. However, in many cases, the central nervous system is required to estimate the external stimulus at much shorter times. What readout mechanism could account for such fast decisions? We suggest a readout mechanism that estimates the external stimulus by the first spike in the population, the tWTA. We show that the tWTA can account for accurate discriminations between a small number of choices. We find that the accuracy of the tWTA is limited by the neuronal baseline firing. We further find that, due to baseline firing, the single first spike does not encode sufficient information for estimating a continuous variable, such as the direction of motion of a visual stimulus, with fine resolution. In such cases, fast and accurate decisions can be obtained by a generalization of the tWTA to a readout that estimates the stimulus by the first n spikes fire by the population, where n is larger than the mean number of baseline spikes in the population.
neuroscience/behavioral neuroscience
neuroscience/sensory systems
neuroscience/motor systems
computational biology/computational neuroscience
neuroscience/natural and synthetic vision
neuroscience/theoretical neuroscience
The Temporal Winner-Take-All Readout