How to read out spatio-temporal spike patterns generated by populations of neurons is fundamental to the understanding of neural network computation. Most of the previous studies on population coding were limited to the static case where only spike countsfor a preset time window are considered. For the encoding of continuously varying signals, however, it is important to understand how the accuracy of population codes is affected by the dynamics of neural spike generation.
Here, we studied dynamic population codes with noisy leaky integrate-and-fire neurons. We presented an algorithm for Bayesian decoding similar to the one presented in Cunningham et al. (2008
). In addition, we derived an approximate algorithm which yields a simple spike-by-spike update rule for recursively improving the stimulus reconstruction whenever a new spike is observed.
The decoding rules can also be applied for decoding the spike trains of populations of neurons, not just single neurons. Importantly, we do not have to assume that the neurons are uncoupled, i.e. conditionally independent given the stimulus. In particular, as we assume the encoding model to be known, we would also know the parameters describing the couplings between neurons. Then, the influence of one spike of a neuron on the membrane potential of any other neuron is just a known, given input and can be subtracted. Therefore, the same decoding framework can also be used for decoding coupled neurons.
The decoding rule is nonlinear and sensitive to the relative latencies between each spike and its predecessor in the population. However, it is not optimal as it does not use the information that the membrane potential stays below threshold between spikes. To incorporate this kind of knowledge one has to integrate the coefficient distribution over the linear halfspace confined by the threshold similar to the method described in Paninski et al. (2004
) but with the additional complication that, the distribution is not Gaussian. Therefore, the optimal Bayesian decoding rule would be computationally much more expensive.
The main goal of this work was to derive a simple decoding rule that facilitates the analysis of neural encoding strategies such as efficient coding, unsupervised learning, or active sampling. Bayesian approaches are particularly useful for these problemsas they do not yield a point estimate only but also aim at estimating the posterior uncertainty over stimuli. Having access to this uncertainty allows one to optimize receptive field properties or other encoding parameters in order to minimize the reconstruction error or to maximize the mutual information between stimulus and neural population response. In this way it becomes possible to extend unsupervised learning models such as independent component analysis (Bell and Sejnowski, 1995
) or sparse coding (Olshausen and Field, 1996
) to the spatio-temporal domain with spiking neural representations. This seems highly desirable as comparisons between theoretically derived models and experimental measurements would thus become feasible.
Furthermore, animals do not receive the sensory input in a passive way but actively tune their sensory organs to acquire the most useful data, for example by changing gaze or by head movements. Such active sampling strategies are related to the theory of optimal design or active learning (Lewi et al., 2009
), where the next measurement is selected in order to minimize the current uncertainty about the signal of interest. Such active sampling strategies give rise to ‘saliency maps’, which encode the expected information gain from any particular stimulus.
Maximizing the mutual information between stimulus and neural response is equivalent to minimizing the posterior entropy. Because of the Gaussian approximation, this can be done in our model by performing a gradient descent on the log-determinant of the posterior covariance matrix. The gradient can be calculated from Eq. 17
. However, the approximated posterior covariance derived in this paper might also be subject to a systematic deviation from the exact covariance matrix. Therefore, an important extension of the present work would be to correct for a bias in the approximate covariance estimate, too. In general the approximations considered in this paper usually tend to over-estimate the true underlying uncertainty, as they wrongly donot cut-off regions in the parameter space.
In this paper, we chose to represent the stimulus by a superposition of a finite set of basis functions as this has some practical advantages. Alternatively, it is also possible to start from a full Gaussian process as stimulus model and then derive a discretization for numerical evaluation. Analogous to the mean vector and covariance matrix of a finite-dimensional normal distribution, a Gaussian process prior over the stimulus is specified by the mean and covariance function of the process. For numerical evaluation it is necessary to choose a grid of time points yielding a finite dimensional normal distribution again. Note that for inference, integrals on the grid points have to be evaluated numerically and therefore a fine time resolution for the si
should be chosen. Therefore, the computational load of decoding a discretized Gaussian process is considerably higher. For practical reasons, we can restrict the inference procedure to a time window around the current spikes, provided that the covariance function falls off quickly. In the non-leaky case with no receptive fields this is the same setting as in Cunningham et al. (2008
The extension to the Gaussian process setting is conceptually important as it allows one to replace the somewhat artificial threshold noise model by membrane potential noise. The dynamics can then be described by a stochastic differential equation. Although the likelihood is much harder to calculate (Paninski et al., 2004
), it still has the renewal property and therefore a similar approximation scheme might be applicable. However, it has the further complication, that the obtained likelihood is only for a given threshold and therefore the threshold has to be marginalized. We hope that more studies will be devoted to the problem of decoding time-varying stimuli from populations of spiking neurons in the future. In particular, it will be crucial to achieve a good trade-off between the basic dynamics of neural spike generation, the accuracy of posterior estimates and the computational complexity of the decoding algorithm.