Spatio-temporal Representations of Uncertainty in Spiking Neural Networks
It has been long argued that, because of inherent ambiguity and noise, the brain needs to represent uncertainty in the form of probability distributions. The neural encoding of such distributions remains however highly controversial. Here we present a novel circuit model for representing multidimensional real-valued distributions using a spike based spatio-temporal code. Our model combines the computational advantages of the currently competing models for probabilistic codes and exhibits realistic neural responses along a variety of classic measures. Furthermore, the model highlights the challenges associated with interpreting neural activity in relation to behavioral uncertainty and points to alternative populationlevel approaches for the experimental validation of distributed representations. Core brain computations, such as sensory perception, have been successfully characterized as probabilistic inference, whereby sensory stimuli are interpreted in terms of the objects or features that gave rise to them [1,2]. The tenet of this Bayesian framework is the idea that the brain represents uncertainty about the world in the form of probability distributions. While this notion seems supported by behavioural evidence, the neural underpinnings of probabilistic computation remain highly debated [1,2]. Different proposals offer different trade-offs between flexibility, i.e. the class of distributions they can represent, and speed, i.e. how fast can the uncertainty be read out from the neural activity. Given these two dimensions, we can divide existing models in two main classes.
The first set, which we will refer to as spatial codes, distributes information about the distribution across neurons; the activity of different neurons reflects different values of an underlying random variable (alternatively, it can be viewed as encoding parameters of the underlying distribution [1, 2]). Linear probabilistic population codes (PPCs) are a popular instance of this class, whereby the log-probability of a random variable can be linearly decoded from the responses of neurons tuned to different values of that variable [3]. This encoding scheme has the advantage of speed, as uncertainty can be decoded in a neurally plausible way from the quasi-instantaneous neural activity, and reproduces aspects of the experimental data. However, these benefits come at the price of flexibility: the class of distributions that the network can represent needs to be highly restricted, otherwise the network size scales exponentially with the number of variables [1].
This limitation has lead to a second class of models, which we will refer to as temporal codes.These use stochastic network dynamics to sample from the target distribution [4,1]. Existing models from this class assume that the activity of each neuron encodes a different random variable; the network explores the state space such that the time spent in any particular state is proportional to its probability under the distribution [4]. This representation is exact in the limit of infinite samples. It has several important computational advantages (e.g. easy marginalization, parameter learning, linear scaling of network size with the number of dimensions) and further accounts for trial-totrial variability in neural responses [1]. These benefits come at the cost of sampling time: a fair representation of the underlying distribution requires pooling over several samples, i.e. integrating neural activity over time. Some have argued that this feature makes sampling unfeasibly slow [2].
Here we show that it is possible to construct spatio-temporal codes that combine the best of both worlds. The core idea is that the network activity evolves through recurrent dynamics such that samples from the posterior distribution can be linearly decoded from the (quasi-)instantaneous neural responses. This distributed representation allows several independent samples to be encoded simultaneously, thus enabling a fast representation of uncertainty that improves over time. Computationally, our model inherits all the benefits of a sampling-based representation, while overcoming potential shortcomings of classic temporal codes. We explored the general implications of the new coding scheme for a simple inference problem and found that the network reproduces many properties of biological neurons, such as tuning, variability, co-variability and their modulation by uncertainty. Nonetheless, these single or pairwise measures provided limited information about the underlying distribution represented by the circuit. In the context of our model, these results argue for using decoding as tool for validating distributed probabilistic codes, an approach which we illustrate with a simple example.
The authors present a recurrent spiking neural network model for sampling from multivariate probability distributions through MCMC. A key advantage of their approach is to separate computation and representation: Samples of the target distribution are obtained from the network response via a linear decoder, i.e. neurons do not need to directly correspond to random variables. This gives rise to a rich distributed spatio-temporal code to represent the underlying distribution. Furthermore, the resulting flexibility in the network architecture allows to run multiple MCMC chains in parallel and, thus, to approximate the target distribution near-instantaneously.The theory is complemented by computer simulations that (a) demonstrate the general feasibility of the approach, (b) illustrate how the network performs inference (in a simple generative model) and (c) compare response characteristics with findings from experimental neuroscience. Finally, the authors explore how the approach can be used to identify (properties of) the underlying distribution when the linear decoder is not known; an important step for an application to experimental data.Overall, this is a good manuscript. It contributes several new ideas which are of interest to a broader research community. The manuscript is mostly very well written (particularly the introduction and discussion), the computer simulations and figures illustrate the network properties well. Finally, the work perfectly fits the scope of the conference.Nonetheless, I have some critical remarks which reduced the otherwise positive impression.(A) The manuscript suffers from missing or extra words which, in some cases, even affect the content. E.g. in lines 46, 47, 142, 153, 188, 223 and 386.(B) The spike trains o_i are not defined in line 80. The reset mechanism (described in [6]) is not mentioned around line 140.(C) The derivation is highly compressed and definitely demands reading of refs [6-8]. Also the SI contains relevant information (e.g. the generative model definition). Furthermore, I didn't find important parameters of the simulations (e.g. number of neurons and decoding matrix in Fig 1, lambda, tau_slow,...). This renders a verification of the results almost impossible. Some details to the simulations should be added to the SI.(D) While the network supports a high degree of freedom in the recurrent weight matrices, connections are still symmetric (at least for the Gaussian case). Such limitations should be mentioned at some point.(E) I was wondering if all equations are fully correct. I didn't delve into the details, so the following are just some points the authors might want the check. In line 141, shouldn't the summation run over the first index of the decoder? The sparseness parameter lambda changes the spike response but does not enter any other equation to account for this effect. Is this correct? Similarly, the time constant tau_slow that scales the recurrent weights and drift is not further specified. Can it be chosen arbitrarily, or is this tau_v?In summary, I had the impression that a bit too many different research questions were put into this nine pages manuscript. This comes at the cost of reduced clarity of the individual parts. For instance, since the derivation rests upon some approximations, it would be helpful to explore the range of validity. Also the intriguing multi-chain sampler would have deserved a more comprehensive presentation. Still, I consider this manuscript an important and valuable contribution to the conference if the authors address some of the above issues, and I hope that (one or more) follow-up papers can investigate the presented ideas in greater detail.