Ensembles of Spiking Neurons with Noise Support Optimal Probabilistic Inference in a Dynamically Changing Environment

It has recently been shown that networks of spiking neurons with noise can emulate simple forms of probabilistic inference through “neural sampling”, i.e., by treating spikes as samples from a probability distribution of network states that is encoded in the network. Deficiencies of the existing model are its reliance on single neurons for sampling from each random variable, and the resulting limitation in representing quickly varying probabilistic information. We show that both deficiencies can be overcome by moving to a biologically more realistic encoding of each salient random variable through the stochastic firing activity of an ensemble of neurons. The resulting model demonstrates that networks of spiking neurons with noise can easily track and carry out basic computational operations on rapidly varying probability distributions, such as the odds of getting rewarded for a specific behavior. We demonstrate the viability of this new approach towards neural coding and computation, which makes use of the inherent parallelism of generic neural circuits, by showing that this model can explain experimentally observed firing activity of cortical neurons for a variety of tasks that require rapid temporal integration of sensory information.

them was independently drawn from a set of ten possible shapes, i.e, s i ∈ {1, . . . , 10}. Based on this information, the monkey had to decide whether to make an eye movement to a red or a green target to obtain reward. A weight w i was assigned to each shape i and kept fixed during the whole experiment. The sum of the four weights associated with the shapes shown in a trial established the probability that reward would accompany one of the two choices [1].
Denoting the time of the presentation of the n-th shape as epoch n, the log-likelihood ratio (measured in units of bans) for an eye movement to the red target for symbols up to epoch n is defined as logLR n = log 10 P (s 1 , . . . , s n |reward at red target) P (s 1 , . . . , s n |reward at green target) .
For a red target, this value is exactly the sum of weights once all four shapes are shown (n = 4), and it is roughly proportional to the sum of the n shapes in the n-th epoch for n < 4. Neurons in LIP were shown to have firing rates proportional to the logLR.
We modeled this task by assuming that P (s 1 , . . . , s 4 |reward at red target) factorizes such that P (s 1 , . . . , s 4 |reward at red target) = 4 i=1 P (s i |reward at red target). As we will see below, this is a very good approximation to the true likelihood. With this assumption, the task falls into task class B (evidence integration). The optimal solution can be approximated by particle filtering for a random variable with two hidden states (rewarded eye movement to the red or green target respectively), see Figure 1A. Each shape was represented by one afferent neuron.
In addition, we introduced afferent neurons that represented the presence of saccade-targets for reasons that are detailed below. See Methods in this supporting text for details on this simulation.   Figure 2c in [1]. In panels B and D, firing rates for identical logLRs were averaged to obtain the data points.
shapes appear ( Figure 1C). This transient is caused by network input which leads to an increase of circuit activity that is compensated subsequently by lateral inhibition. The first transient is larger due to additional input from afferents that represent saccade targets. Hence, the strong response in the first epoch seen in experiments is explained in our model by transient responses to additional circuit input.

Methods
We simulated a particle filter circuit with ensemble size M = 1000 and estimation sample size L = 400; The lateral inhibition scaling was set to I lat 0 = 0.25. No action readout layer was added.
12 evidence neurons provided input to the circuit. Two of them represented the saccade targets. Each of the remaining eight neurons represented one of the eight shapes that determined the probability of reward. Spikes of the afferent neurons were produced as follows. First, the sequence of shapes s 1 , . . . , s 4 that was presented was drawn randomly. All possible sequences were equally probable. As in the experiment by Yang and Shadlen [1], the two 'trump' symbols were not presented. After 100ms waiting time, the two targets for the eye-movement and the first shape appeared. This caused a spike in the afferent neurons for the two targets and the afferent neuron for s 1 at a time that was uniformly distributed in [100,110]ms. Subsequently, the afferent neuron for s 1 continued to spike in a Poissonian manner with rate f in = 10Hz.
The other three targets appeared at respective times 250ms, 400ms, 550ms and spikes in the corresponding afferent neurons were produced analogously.
To determine weights for connections from evidence neurons to neurons in the evidence layer, we used λ jl = P (s i = j|a = l)10Hz. Here, P (s i = j|a = l) is the probability that shape j is observed in epoch i, given that the rewarded action is l. Note that this probability is the same for all epochs i. It can be computed as follows: We obtain the joint via marginalization: where the sum runs over all shape sequences where the first shape is j. The conditional for movement to the red target P (a = R|s 1 , s 2 , s 3 , s 4 ) is given by P (a = R|s 1 , s 2 , s 3 , s 4 ) = 10 P i w i 1 + 10 P i w i , and the conditional for the green target is P (a = G|s 1 , s 2 , s 3 , s 4 ) = 1 − P (a = R|s 1 , s 2 , s 3 , s 4 ) [1]. Here, w i is the weight associated with symbol s i that was set to w = (−70, −0.9, −0.7, −0.5, −0.3, 0.3, 0.5, 0.7, 0.9, 70). For the afferent neurons indicating the targets for saccades, λ ji was set to 0.8Hz for both states i.