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

<div><p>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.</p></div

Particle filter circuit architecture for task classes B and C.

<p><b>A</b>) Circuit with ensembles (indicated by red and blue neurons respectively) and neurons per ensemble. Neurons in layer receive synaptic connections from neurons in layer and update the represented distribution according to evidence input from afferent neurons (green). Lateral inhibition (magenta; see panel C) stabilizes activity in this layer. Neurons project back to layer . For task class B (evidence integration; static random variable ), only connections between neurons that code for the same hidden state are necessary and layer simply copies the distribution represented by layer , see <i>Task class A</i> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g003" target="_blank">Figure 3</a> (in contrast to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g003" target="_blank">Figure 3</a>, the copying ensembles are plotted above ensembles in order to avoid a cluttered diagram). For task class C (Bayesian filtering; random variable with time-independent dynamics), implements changes of the represented distribution due to the dynamics of the random variable and is potentially fully connected to . Neurons in layer disinhibit neurons in layer (double-dot connections; see panel B). Disinhbition and lateral inhibition is indicated by shortcuts as defined in B, C. Arrows indicate efferent connections. A schematic overview of the circuit is shown in the inset. <b>B</b>) Disinhibition : neurons excite an interneuron (purple) which inhibits the inhibitory drive to some neuron . As a graphical shortcut, we draw such disinhibitory influence as a connection with two circles (inset) <b>C</b>) Lateral inhibition: Pyramidal cells (blue) excite a pool of inhibitory neurons (magenta) which feed back common inhibition . The graphical shortcut for lateral inhibition is shown in the inset.</p

Particle filtering in ENS coding for the ambiguous target task.

<p><b>A</b>) Represented random variables. <b>Aa</b>) Evidence integration is performed for a random variable with 16 hidden states corresponding to direction-color pairs. Values of the random variable are depicted as circles. Observations accessible to the monkey in one example state are shown as boxes. <b>Ab</b>) The action readout layer infers a color-independent random variable by marginalization over color in each direction. <b>B</b>) Circuit structure. The circuit on the top approximates evidence integration through particle filtering (top gray box; : dynamics layer ensembles; : evidence layer ensembles)) on the random variable indicated in panel (Aa). An action readout layer (bottom gray box; ensembles <i>X</i>) receives feed-forward projections from the particle filter circuit. <b>C</b>) Spike rasters from simulations for afferent neurons (Ca) and neurons in the action readout layer (Cb). Each line corresponds to the output of one neuron. Afferent neurons are ordered by feature selectivity (e.g., top neurons code the presence of the fixation cross). Action readout neurons are ordered by preferred movement direction. See also <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g001" target="_blank">Figure 1</a>.</p

Notation.

<p>Description of frequently used variables for easy reference. In general, capital letters refer to ensembles and lower case letters to neurons in these ensembles.</p><p>Notation.</p

Particle filter circuit architecture for task class D.

<p>Extended circuit with ensembles (indicated by red and blue neurons respectively) and neurons per ensemble and two possible contexts. Ensembles in layer are duplicated for each context. These neurons receive context information via disinhibition from context neurons (yellow; only connections from context 1 shown for clarity). Disinhbition and lateral inhibition indicated by shortcuts as defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g004" target="_blank">Figure 4B, C</a>. Arrows indicate efferent connections. A schematic overview of the circuit is shown in the inset.</p

Evidence integration through particle filtering in ENS coding.

<p><b>A</b>) The state of a binary random variable that gives rise to two possible observations is estimated. Both observations occur more frequently in state 1 (indicated by sharpness of arrows). <b>B</b>) Estimation is performed by a particle filtering circuit with evidence input (: dynamics layer ensembles; : evidence layer ensembles). <b>C</b>) An evidence spike is observed at times 20 ms and 25 ms in evidence neuron and respectively. <b>D</b>) Example for the rate dynamics in layer . Ensemble rate for ensemble (black) and whole layer (gray). The input leads to a transient increase in the ensemble rate. Inhibition recovers baseline activity. The ensemble rate for state 1 undergoes a transient and a sustained activity increase. <b>E</b>) Temporal evolution of estimated posterior probability for state 1 (black) in comparison to true posterior (gray) for this example run. <b>F</b>) Posterior probability at <i>t</i> = 45 ms () for state 1 of the circuit in comparison to true posterior at this time (). Each dot represents one out of 100 runs with prior probabilities and observation likelihoods drawn independently in each run (see <i><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#s4" target="_blank">Methods</a></i>). The results of the example run from panels A–E is indicated by a cross.</p

Tracking of dynamics in ENS coding.

<p><b>A</b>) The state of a binary random variable is estimated where state 1 transitions to state 2 with some transition rate . <b>B</b>) Estimation is performed by a particle filter circuit without evidence input (: dynamics layer ensembles; : evidence layer ensembles). <b>C</b>) Example for the rate dynamics in layer . Ensemble rate for ensembles (black) and whole layer (gray). While rates in ensembles change due to the prediction of a transition, inhibition keeps the overall firing rate in the layer approximately constant. <b>D</b>) Temporal evolution of estimated posterior probability for state 2 (black) and true posterior (gray) for this example run. <b>E</b>) Circuit estimates of posterior probabilities at time <i>t</i> = 50 ms () in comparison to true posteriors at this time (). Shown are 100 runs (dots) with prior probability for state 1 and transition rate drawn from uniform distributions in [0.1, 0.9] and [0, 30]Hz respectively in each run. The result of the example run from panels C-D is indicated by a cross.</p

Particle filter circuit equations for task class D.

<p>Here we have defined and . denotes lateral inhibition and disinhibition. is an arbitrary constant.</p><p>Particle filter circuit equations for task class D.</p

Computations in ENS coding in a feed forward circuit architecture.

<p>A binary random variable is represented in ENS coding through neurons . The posterior for a binary variable is represented by neurons . Each variable is represented by ensembles, one for each possible state (indicated by neuron color), and neurons per ensemble. The two layers are connected in an all-to-all manner. Arrows indicate efferent connections (i.e., outputs in ENS coding). The architecture is summarized in the inset.</p

Context-dependent Bayesian filtering in two successive trials of the ambiguous target task.

<p><b>A</b>) Represented random variables. <b>Aa</b>) Dynamics of a random variable that codes the current phase in a trial of the ambiguous target task (CHT: fixation; SC: spatial cue; MEM: memory cue; CC: color cue). Possible observations in each phase are indicated in boxes. <b>Ab</b>) Context-dependent Bayesian filtering is performed for a random variable with 16 hidden states corresponding to direction-color pairs as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g006" target="_blank">Figure 6A</a>a. Gray lines indicate context-dependent transitions. All-to-all transitions are possible in the fixation phase (CHT). There are no transitions in other phases of a trial. <b>Ac</b>) The action readout layer infers a color-independent random variable by marginalization over color in each direction. <b>B</b>) Circuit structure. The circuit on the top (ensembles and ) performs Bayesian filtering on the random variable indicated in panel (Aa). It provides context for another particle filter circuit (middle gray box; ensembles and ) that generates a belief about the random variable indicated in Ab. An action readout layer is added (bottom gray box; ensembles ). <b>C</b>) Spike rasters from a simulation of two successive trials for afferent neurons (Ca), neurons in the particle filter circuit for the phase in the trial (Cb), and neurons in the action readout layer (Cc). Neurons in Cb are coding for the current phase of the trial (ordered from bottom to top: CHT, SC, MEM, and CC). Neuron ordering in Ca and Cc as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003859#pcbi-1003859-g006" target="_blank">Figure 6</a>.</p