Background: Balanced spiking network model

Here we provide a brief introduction to the original balanced spiking network (BSN) framework introduced by Boerlin, Machens, & Denève [31]. The goal is to design a spiking network that can accurately implement an arbitrary linear dynamical system. Consider a linear dynamical system defined by: (1) where x(t) = (x₁(t)…x_J(t))^⊤ is a vector of J dynamic variables that we will refer to as the target, A is the J × J linear dynamics matrix, and c_t = (c_i(t), …c_J(t))^⊤ is a J-dimensional vector of inputs. The BSN model consists of a spiking network of N neurons that attempts to approximate the target output x(t) via a weighted combination of filtered spike trains: (2) where r(t) is the set of spike trains convolved with an exponential decay function, and W are J × N readout weights. (See Fig 1 for a schematic.) In general, for a 1-D dynamical system, the population is divided into equal pools of ‘positive’ and ‘negative’ neurons (depending on the signs of their individual weight components) although this is not a strict requirement. For J > 1, the ‘positive’ vs ‘negative’ distinction does not necessarily apply as the signs of the weights need not be consistent across dimensions.

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Fig 1. A diagram illustrating the BSN.

Neurons receive stimulus input projected onto the transpose of a set of linear weights, W^⊤, and the output is reconstructed by filtering spikes through the same weights, W. Neurons are connected via two coupling weights: fast synapses, W^⊤W, which instantaneously propagate individual spikes through the network, and slow synapses, , which implement network dynamics by feeding the filtered spike trains back into all neurons in the network. The network is divided into two equal populations of positive (red) and negative (blue) output weights, whose spikes have opposite effects on network output. Self-connections for these neurons are shaded in their respective colors, for visualization, but are always negative.

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The i’th component of the vector r(t) is given by (3) where denotes the i’th neuron’s spike train, defined by a series of delta functions at spike times , and τ is the time constant of the exponential filter h(t).

From this starting assumption, Boerlin et al introduce a greedy update rule that causes a neuron to spike whenever doing so will reduce the squared error between target x(t) and network output , (4) which is mathematically equivalent to the threshold-crossing spiking rule in a leaky integrate-and-fire (LIF) neuron.

Here we recapitulate the derivation of this spiking rule in discrete time, for clarity and ease of implementation. Let denote the network output at time bin t. The effect of adding a spike from neuron i in this time bin would be to augment the output vector by that neuron’s decoding weight vector w_i, which is given by i’th row of the decoding weight matrix W. Thus, network output is if neuron i is silent and if it spikes. This suggests that the neuron should spike if doing so will result in smaller error, or (5)

Simplifying this expression yields the condition that the neuron should spike if projection of the error vector onto w_i is greater than half the squared L₂ norm of w_i: (6)

Boerlin et al therefore suggest regarding the time-dependent left hand side of (Eq 6) as the membrane potential for neuron i, and the right hand side as its spike threshold T_i: (7) (8)

Under this view, each neuron is computing a local approximation to the difference between the true target x_t and the network’s current output , projected onto that neuron’s weight vector w_i.

The only missing piece from this expression is that the neurons do not of course have access to the true value of x_t. But they do have implicit access to A, and thus to the dynamics governing x_t. Boerlin et al therefore propose that the the network output is sufficiently close to the target output x_t that it can be used to accurately approximate the desired dynamics: . To make this explicit, we introduce a proxy variable z_t, which denotes the network’s own (internal) approximation to the true target x_t. This proxy variable evolves according to (9) where for simplicity we use a forward Euler method for integrating the dynamics equation (Eq 1) with time bin size Δ. Higher accuracy can be achieved using exponential Euler integration (see Methods). Of course z_t is never represented explicitly; the network tracks z_t via its projection onto the decoding weights W, as we will see shortly.

Limitations of the BSN model

A key limitation of the BSN model is that it requires unrealistically fast communication between neurons. In the standard integrate-and-fire model, a spike resets only the membrane potential of the neuron that emitted it. In the BSN model, by contrast, the membrane potentials of all neurons reset following a spike from any neuron via the −W^⊤Ws_t term in (Eq 16). The instantaneous reset following a spike is necessary to ensure that each neuron’s membrane potential maintains an accurate representation of the read-out after each spike. From a normative standpoint, the hard LIF threshold entails that maintaining an accurate local copy of the error is critical for the network to encode the target.

In fact, the problem is slightly worse than this: standard implementations of the BSN model include a constraint that only one neuron is allowed to spike in a single time bin as a way of imposing instantaneous (i.e., sub-time bin) communication. Without this constraint, neurons with similar output weights tend to spike in the same time bin, when a spike from any one of them would have sufficed to compensate for error in network output. This causes the network output to dramatically overshoot the target. In the subsequent time bin, neurons with opposite-sign weights fire to compensate for this error, and overshoot the target by a large amount in the opposite direction. This sets up a pathological pattern of oscillatory firing known as “ping-ponging”, in which two populations spike maximally in alternating time bins of the discrete simulation [31, 36].

Fig 2 illustrates how ping-pong behavior can arise if multiple spikes are allowed in a single time bin. We set the BSN model to implement a 1-dimensional exact integrator, , using the same parameters as the example from figure 1C of [31]. In brief, the network contained 400 neurons, divided into two equal sized populations with output weights of +0.1 and -0.1, respectively, which we refer to as positive-output and negative-output neurons (see Methods for complete details). When the rule forbidding multiple spikes per time bin is imposed, the network accurately tracks the target output variable (Fig 2A). However, removing this constraint—allowing all neurons with membrane potential above threshold to fire—results in ping-pong behavior and large errors in tracking the target (Fig 2B).

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Fig 2. Balanced spiking network implementing an exact integrator.

The network consists of 400 neurons, divided into two populations with output weights of + 0.1 (red) and −0.1 (blue). (A) Simulation results under the condition that only one neuron is allowed to fire per discrete time bin. (B) Simulation results when all neurons whose membrane potential is above threshold in a single time bin are allowed to fire, leading to “ping-pong” behavior. Insets show that the read-out (yellow) is alternating between large over- and under-estimates of the target (in black). Insets show, in order from top to bottom: the voltage traces of neurons in both positively and negatively weighted populations for a small time window, the resulting spikes in each time bin, and the resulting read-out (yellow) and target (black). Since the weights and inputs are identical across populations, so are the voltage traces. Ping-ponging results, as all neurons within a population cross the threshold in the same time bin, spike, and cause the read-out to oscillate between over- and under-estimates of the target.

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Note that ping-ponging could be eliminated by computing threshold-crossing times with extremely high temporal precision, either using very small time bins or spike-time interpolation methods [37, 38]. Such precision may allow us to identify the first neuron to spike and to near-instantaneously reset the membrane potential of other neurons, preventing overshoot of the target. However, this near-instantaneous reset of the membrane potential in other neurons is at odds with the timescale of synaptic communication, and represents a shortcoming of the original BSN framework that our work seeks to overcome. (See Discussion for an overview of other approaches to the problem of instantaneous synapses, e.g. [36]). In the following sections, we imposed the constraint that only one neuron could spike per time bin for all simulations of the original BSN model, consistent with [31].

BSN with conditionally Poisson neurons

To overcome the problems of instantaneous transmission and network instability, we propose two novel formulations of balanced spiking networks with conditionally Poisson neurons: (1) a local framework, where each neuron spikes independently based on its local estimate of network error; and (2) a population-level framework, which sets firing rates to reduce expected error for the entire population.

The key idea in both frameworks is to replace the deterministic integrate-and-fire spiking rule with probabilistic spiking. Under this modified spiking rule, spiking is governed by an instantaneous probability of spiking λ_t, also known as the conditional intensity, such that spiking is independent with probability Δλ_t in any small time window of width Δ. This results in an auto-regressive Poisson generalized linear model (GLM), also known as a Cox process [33–35]. This model has a quasi-realistic biophysical interpretation [32, 39, 40], and recent work has shown that it can capture a wide range of dynamical behaviors found in real neurons [35].

Local framework.

A simple way to introduce probabilistic spiking to the BSN framework is to replace the hard spike threshold of the integrate-and-fire model with a soft threshold, so that spike probability grows as a nonlinear function of membrane potential, an approach also known as the “escape-rate approximation” [32]. Specifically, we define each neuron’s conditional intensity function to be a sigmoidal function of membrane potential: (17) where v_t is the membrane potential at time t, T is the spike threshold, α is a slope parameter governing the sharpness of the threshold, F_max is the maximal firing rate and F_min is a baseline firing rate, meant to simulate random firing activity in the absence of a stimulus. The probability of spiking in a small time interval is proportional to λ_t, which models the spike response as an inhomogenous Poisson process. We refer to this as the local Poisson framework because, like the BSN, spikes are generated by internal dynamics that evolve according to local copies of the representation error. See Fig 3 for a schematic.

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Fig 3. Schematic of two neurons in a BSN with conditionally Poisson neurons.

The stimulus influences each neuron’s membrane potential v_i via a set of input weights W^⊤. The neurons reset themselves via instantaneous, fast synapses. Fast connections to other neurons propagate the effects of spikes with a synaptic time delay d. The desired linear dynamics are implemented via slow weights (through spike trains filtered by an exponential) also with a time delay d. Within each neuron, spiking is probabilistic with an instantaneous probability of firing λ_i(t) = f(v_i(t)), where f(⋅) is a nonlinear function of voltage. Self-connections are only shown for neuron i.

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Although it is common to use exponential nonlinearities for Poisson GLMs, here we have used a scaled sigmoid function to control both the suddenness of firing onset and the maximum achievable firing rate. The parameter α controls the precision of firing onset, while F_max and F_min control the range of firing rates within a small time window. The resulting function (Eq 17) resembles an exponential function at low firing rates but saturates at a maximum of F_max (see Fig 4). If we let both α, F_max → ∞, we recover the (hard-threshold) integrate-and-fire rule of the original BSN, in which a spike occurs probability 1 when v_t > T. However, for finite α and F_max, the onset of spiking is more gradual.

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Fig 4.

(A) The conditional intensity for the exponential non-linearity (dashed lines) and the sigmoid non-linearity (solid lines). The conditional intensity of the sigmoidal non-linearity closely follows that of the exponential non-linearity for sub-threshold voltages, but levels off after threshold, keeping firing rates stable. (B) Family of nonlinearities with varying F_max. Increasing F_max raises the firing rate at which the nonlinearity saturates. (C) Family of nonlinearities with varying α. Increasing α increases the steepness of the nonlinearity, which approaches a hard-threshold function as α → ∞ (like the BSN). (D) Simulation of the original BSN implementing an exact integrator, showing membrane potential and spikes of a single example neuron. (E) Spikes and membrane potential of the same neuron in a local Poisson BSN implementation of the same system. High α simulations (yellow) replicate the behavior of the BSN integrator. Lowering α to 50 (blue) or 10 (red) results in a spread of spikes centered around the deterministic BSN spikes.

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As a practical matter, we do not wish to allow a single neuron to fire multiple spikes in a single time bin, because the first spike would preclude additional spikes in the same time bin due to “reset” of the membrane potential. We therefore simulate the model with the spiking rule: (18) where exp(−Δλ_t) is the probability of observing no spikes in a time bin of size Δ under the Poisson model. For each time-step, we update v as in the original BSN model, pass it through the nonlinear function f(⋅) to obtain the vector of Poisson firing rates, λ_t, and draw spikes as independent Bernoulli random variables with probability as given above.

Fig 4 illustrates how α affects spiking precision by comparing spike times of a single neuron integrating a noisy stimulus implementing the original BSN framework (Fig 4D) to the local framework with different values of α (Fig 4E). As we expect, for high values of α we recover the precise spiking behavior of the BSN model. As α decreases, spike times spread around the ideal BSN spikes. Note that the precise amount of spikes fired in the time window (four) is conserved, although the timing is highly variable. On average, though, the local framework has spike times centered around those of the original BSN.

Robustness to parameters of the nonlinearity.

We studied the effects of varying α, F_max, and F_min on the performance of the homogeneous integrator network (Fig 5) and found that there exists a wide range of values for which the network error is low and the spiking activity is efficient (Fig 5B and 5C). The rightmost panel on Fig 5A shows raster plots and corresponding read-outs for ‘optimal’ parameter settings (defined as being in the low error and activity range, denoted by the red * in Fig 5B and 5C), and in the subsequent panels we modify each of the parameters in turn to show qualitative changes in spiking activity and read-out accuracy. Decreasing α and F_max negatively impacts read-out quality, while removing background spiking returns the network to a high-precision, synchronized regime.

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Fig 5.

(A) Simulations of local Poisson model showing the effects of varying the parameters of the soft-threshold nonlinearity on performance. Relevant parameters are the slope α, maximal firing rate F_max, and baseline firing rate F_min. Network dynamics implemented an exact 1D integrator and the stimulus was the same as Fig 2. Red and blue dots indicate spikes from neurons with positive and negative output weights, respectively. (B) Network performance as quantified by R² across a range of parameter settings with baseline fixed at F_min = 0 (log-scale). Red asterisk indicates the values for the rightmost column of A (α = 1000, F_max = 100). Accuracy remains high across a broad range of parameter values, falling substantially below 1 when slope and maximum firing rate are both large or very low. (C) Percent of the neural population active as a function of α and F_max, with F_m in = 0. The network shows ping-ponging behavior in upper right corner, where the model approaches a deterministic, hard-threshold firing rule.

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As shown in figure (Fig 5B), network performance noticeably deteriorates for very large values of α and F_max. This happens because we are forcing the precision to be too high through α while allowing neurons to spike too frequently through F_max. Since the spiking rule is localized, large proportions of the population are active at the same time (Fig 5C), much what happens when the original BSN starts to ping-pong. However, the values at which this happens (e.g., F_max > 10³ spikes/second) are well above what we would expect to see in a biological system. Likewise, we observe a low R² value when α and F_max are very low. In this case, the probability of spiking is simply too small for spikes to be fired in a given time bin, despite large coding errors. Although these results are using relatively simple target dynamics, we still observe a wide ranges of stable F_max and α settings for more complex or multi-dimensional simulations.

The computational advantage of our probabilistic spiking rule is that it introduces uncertainty into spike timing, therefore preventing all neurons from firing at once. The parameters controlling this asynchrony, F_max, F_min and α, have direct physical interpretations (maximal and minimal/background firing rate and error tolerance, respectively) which can be mapped on to characteristics of real neural circuits.

In theory, a BSN network with voltages corrupted by additive Gaussian noise could behave similarly to the local framework. However, we found that this model still tended to exhibit ping-ponging unless noise amplitude approached the amplitude of signal-induced fluctuations in membrane potential, which adversely affected coding accuracy. By contrast, the local framework is stable even when the resets of other neurons is delayed to the next time bin (see Figs 5 and 6) or by several milliseconds (Fig 7). Intuitively, the reason for this is that, in addition to the slope parameter alpha, which controls the steepness of the soft-threshold-crossing process (in a way, similar to additive voltage noise in the original BSN), we have a limit on the maximal firing rate F_max, which ensures that there will not be too many spikes across the population even when many neurons are driven simultaneously across their (noisy) threshold. That is, it would take several time bins for them all to fire given the value of F_max, which is enough time for synaptic inhibition to arrive and prevent firing. Moreover, in Fig 5B we show that there is a broad range of parameters over which the local population is able to achieve stable, accurate coding. Strictly speaking, by introducing extra degrees of freedom, we are losing the strict normative angle of the original BSN framework. However, what we gain in the process—a considerably expanded space of stable network configurations—makes it possible to introduce realistic communication delays between neurons.

Population framework

We now describe a second framework for implementing BSNs with conditionally Poisson neurons. In this approach, we take a population-level instead of a neuron-level view of the optimization problem to be solved. Instead of assigning each neuron to carry an independent representation of the error between the target and actual network output, we assign each neuron an analog probability of firing such that expected number of spikes across the network compensates appropriately for the total error. We refer to this as the population framework.

The derivation of this framework starts from the same foundation of the original BSN, namely, an error function describing the discrepancy between target and actual network output. However, instead of specifying that each neuron should spike whenever doing so will reduce error (Eq 5), we compute a vector of spike rates λ, such that the expected spike response across the population over some time window of length κ will minimize error. This leads to the following network objective function: (19) where x_t is the target output at time t, is the actual network output at time t, W are the decoding weights, and λ is the vector of firing rates (conditional intensities) of a network of Poisson spiking neurons. In this expression, κW λ is expected contribution to network output over a time window of size κ. For implementation in discrete time, κ should be an integer multiple of the bin size Δ.

To minimize the above error, we set the instantaneous spike rate vector equal to the least-squares solution: (20) where is the Moore-Penrose pseudo-inverse of the decoding weight matrix W. Poisson neurons firing independently with conditional intensity λ_t will therefore minimize the expected error between target and actual network output. Note that if W has orthogonal unit-vector rows, such that WW^⊤ = I, then and we obtain the same encoding weights as the original BSN framework. For the implementation of the population framework, we make the same substitution of the proxy variable z for x.

In the local framework, increasing the population size means more neurons are competing to reduce the read-out error in a single time window. This increased activity can lead to ping-ponging. The population level view of the problem scales the probability of spiking, λ_[i], by WW^⊤, which increases with population size. This re-scaling effectively spreads the ‘responsibility’ of correcting an error across the entire population by incorporating information about the total network size in an individual neuron’s activity.

However, the solution in (Eq 19) is not valid generally because the right-hand-side can take on negative values, whereas the conditional intensity for a Poisson process must be positive. To overcome this, we create two mirrored copies of the population. Positive firing rates are assigned to one copy with weight vector W, and negative firing rates are assigned to the other copy with weight vector −W. We distinguish between these neurons and ‘anti-neurons’ by the sign of their membrane potential as determined by the least squared solution. Similarly to the local and BSN models, spikes from either population will have opposite effects on the output variable. However, in this case these designations are not fixed labels and do not apply to the actual sign of w_i. For example, spikes from the i’th neuron will contribute w_i to the network output, while a spike from its ‘anti-neuron’ counterpart will have a contribution of −w_i to network output, but the entries of w_i themselves may be positive or negative.

Formally, we define the population framework in terms of the update equations: (21) (22) (23) (24) (25) (26) where are the coupling weights from r_t−1 to the pre-spike membrane potentials. This differs from the standard BSN framework in that spikes, rather than being driven by deterministic threshold crossing, arise from a Poisson process with conditional intensity or . If v_[i]t is negative, then is set to zero, and the corresponding anti-neuron’s firing rate is positive. The voltage updates and spiking resets are identical to the local and BSN models.

Fig 6 shows a comparison of the activity of the local and population-level frameworks, as well as the accuracy and spike counts between all three models. Fig 6A and 6B show that the local and population models perform similarly, although the population framework has a higher spike rate. Between the three models, the original BSN model has a lower decoding error than the local and population Poisson models for both the one-dimensional and two-dimensional dynamical systems (Fig 6C), although at the expense of a greatly increased number of spikes (Fig 6D). This is due to the deterministic spike rule, which enforces that any spike fired must perfectly compensate for the error in every given time bin (or as well as is allowed by the size of the decoding weights, W). Spiking in the local and population frameworks is driven by this requirement, but is stochastic.

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Fig 6. Simulations of the local and population frameworks implementing a 1D and 2D dynamical system.

(A) The target was a 1-dimensional integrator: . Left side shows spikes and outputs from local Poisson model, while right side shows spikes and outputs for population Poisson model. As in previous figures, red dots indicate spikes from neurons with positive output weights, blue dots indicate spikes from neurons with negative weights. (B) The target was a 2-dimensional oscillator . For the population model, the time window for computing expected spike count was κ = 5ms (50 time bins). Weights were randomized to be positive or negative in either dimension, such that neurons are no longer divided into strictly positive- or negative-weight groups. (C) Accuracy (measured by root-mean-squared error) of the two models for 1D and 2D systems. (D) Number of spikes emitted by each model during simulations (log scale).

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However, in terms of accuracy, all three models had similar R² values: 0.9961 (BSN), 0.9957 (local), and 0.9928 (population) for the 1-D and 0.9686 (BSN), 0.9395 (local) and 0.9565 (population) for the 2-D dynamical system. Thus, the probabilistic frameworks, on average, compensate for the error as well as the original BSN. But in any given time bin, there is a certain probability of firing too early or too late to optimally compensate for the error, which results in a higher RMSE. Note that the performance of the local and population models depends on the parameters of the firing rate nonlinearity and on κ, respectively, and modifying those can move the models into lower-error (and higher spike count) regimes.

Incorporating synaptic time delays

The original BSN model relies on near-instantaneous synaptic communication between neurons since all neurons in the population must reset immediately after a spike in any neuron. A more realistic model would require that synaptic inputs arrive only after a brief synaptic delay; only the reset of a neuron’s own membrane potential following a spike could be considered instantaneous.

To test the robustness of the two Poisson BSN frameworks introduced above, we altered synaptic currents to incorporate a synaptic delay in neural outputs. In the revised model, filtered spike trains and “fast” membrane potential resets are received by other neurons only after a synaptic delay d. Thus, if neuron fires at time t, it resets its own membrane potential in the next time bin, but we will update spike trains and filtered spike trains received by other neurons only at time t + d.

To compensate for synaptic delays, we altered the network dynamics so that membrane potential—instead of reflecting the current error, as in the original BSN— reflects the network error extrapolated d time steps into the future. The basic logic of this approach is that the network dynamics should look into the future to predict whether firing a spike now will reduce error at the time when the spike actually arrives. The resulting solution is equivalent to minimizing an objective function (Eqs 4 or 19) where x_t+d and take the place of x_t and .: (27) Here, and represent the extrapolated target and network outputs at time t + d, respectively. This solution arises by analytically solving the differential equation , which describes the target dynamics, for z at time t + d given an initial condition z_t and c = c(t), and solving the equation , which describes passive network dynamics in the absence of spiking, for r at time t + d given an initial condition r_t. For the population-level Poisson framework, the encoding weights replace W^⊤ in (Eq 27). (See Methods for details).

Fig 7 shows an analysis of the accuracy of the local and population-level Poisson frameworks with synaptic delays. For both models, a 5ms synaptic delay does not have pronounced effects on the spiking activity or the quality of the read-out (Fig 7A and 7B). Fig 7D shows the R² values as a function of time delay for both frameworks. We compare it against a theoretical upper bound on the coding accuracy (in black), which is a consequence of the exponential Euler approximation (see Methods for details).

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Fig 7. Illustration of local and population conditionally Poisson BSN frameworks with synaptic delays.

(A) Spike trains simulated from the local Poisson framework implementing a 1D exact integrator, both without (top) and with a 1-ms synaptic delay (middle). The network output accurately tracked the target variable for both models (bottom). As before, red/blue spike trains indicate neurons with positive/output weights. (B) Analogous plots for population Poisson framework. (C) Stimulus used for simulations shown in A and B. (D) Coefficient of determination (R²) computed using 50 simulations of each framework. Black trace indicates the maximum possible R² value that could be obtained given the exponential Euler integration rule (see Methods).

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For the local framework, the R² value is lower than for the simulation shown in Fig 6 because the parameters were chosen to make the network more robust to synaptic delays. Otherwise, in the high-precision (high α) regime with synaptic delays, ping-ponging may result, as shown by the higher error regime in Fig 5B. By contrast, the population framework can maintain high levels of accuracy for a large range of synaptic delays.

These revised dynamics could also be used to increase accuracy of a standard BSN with synaptic delays, for example as defined in [36]. We attempted to incorporate synaptic delays into our implementation of the standard BSN, but it led to ping-ponging, after which the model was no longer able to track the target dynamics. The resulting R² values were negative, so they are not included in Fig 7D.

Network performance

Lastly, we analyzed the performance of our local and population frameworks as compared to the original BSN model. Fig 8 shows the cross-correlations of spike trains generated by a network of forty neurons implementing a one-dimensional integrator, , with a white-noise stimulus c(t), for the local, population and original BSN models. Local and population Poisson BSN models both enforced a synaptic delay of 1 ms.

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Fig 8. Cross- and auto-correlations for the original BSN, local Poisson and population Poisson BSN models with synaptic delay.

The top row shows average auto-correlations across both populations of neurons. The bottom row shows average cross-correlations for pairs of neurons with the same sign output weight (i.e., both positive or both negative, in purple) and for pairs of neurons with opposite-sign output weights (e.g., one positive and one negative neuron, in yellow). The original BSN network exhibits negative correlations between neurons with the same sign, and positive correlations between neurons with opposite sign. The local and Population Poisson models show the opposite pattern, which more closely resembles correlations found in neural populations in (e.g.) visual cortex.

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To compute cross-correlations, we divided the neurons into positive-output and negative-output groups. We then computed average within-group (positive-positive and negative-negative) and across-group (positive-negative) cross-correlations. These curves show substantial differences between the original BSN model and the two Poisson models. First, cross-correlations of the original BSN model are 0 at lag zero, due to the rule that only one neuron can spike in a single time bin. More importantly, the within-group correlations for the BSN model exhibit a trough at time zero, meaning that neurons with the same output weight are anti-correlated. Conversely, across-group correlations exhibit an increase at small lags, meaning that neurons with opposite sign output weights are more likely to fire together in a small time window. This relationship is at odds with correlations in both retina and visual cortex, where studies have reported that correlations are highest for neurons with similar tuning, and lowest for neurons with dissimilar tuning [34, 41–43]. By contrast, the local and population Poisson models successfully recapitulate this pattern of correlations, with a peak in the cross-correlations between pairs of neurons with the same sign weights, and a trough for pairs of opposite-sign neurons. Cross-correlations from these models also exhibit no trough at zero due to the lack of a rule prohibiting simultaneous spiking. Thus, cross-correlations represent an additional dimension of biological plausibility of the proposed Poisson frameworks.

Fig 9A shows the relationship between root-mean-square error (RMSE) and network size (N). The RMSE values shown are relative values, normalized in order to best compare the trend in error with respect to N. The original BSN model has a decoding error that scales with , provided that the weights are likewise scaled by , and we find similar scaling for the local and population frameworks, although the BSN has a tighter bound. Poisson rate codes typically scale as due to the Cramér-Rao bound, which places a lower bound on the variance (and therefore the mean-squared-error) of an unbiased estimator. We might expect the local and population networks to scale similarly.

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Fig 9. Scaling of error and spike count with population size (weights held fixed).

(A) Relative root-mean-square error (RMSE) decreases approximately linearly with the network size for all three models. R² values for the fit to 1/N were .98, .91, and .82 for the BSN, local and population models, respectively. (B) Total spike count as a function of network size. The results shown are the average of five simulations of the networks performing exact integration of a signal formed by a sum of two sinusoids.

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However, the scaling applies under the assumption of a constant firing rate. Our local and population models have rates that are a function of voltage dynamics, or the coding error. Although the spiking mechanism is Poisson, the conditional intensity is still driven by the coding error. So even if the Poisson spikes over- or under-shoot the target in a particular time window, on average the network corrects it in the following bins. In sum, this makes our frameworks’ coding capabilities scale better than .

We also looked at how spike counts varied with network size, shown in Fig 9B. In general, as the weights get smaller, the read-out is more precise, but more spikes are needed to reconstruct the target. The local framework initially has a low spike rate because of the large weights and higher error tolerance (α = 1000). However, as the weights get smaller, spiking increases and the local framework exhibits ping-ponging at very small weight sizes. Likewise in the BSN, for small weight sizes the decoding error increases because weights become too small to encode the target given the one-spike rule.

Finally, we looked at how robust both frameworks are to neuron loss. The authors of [31] show in Fig 7G that the original BSN is robust to sudden inactivation of neurons. We ran a similar simulation, shown in Fig 10, silencing 50% of the negatively weighted and positively weighted neurons for.6s at a time by setting the probability of spiking in that time bin to 0. Like in the original BSN, our networks maintain coding accuracy despite large neuron “loss” by increasing the firing rates of remaining neurons.

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Fig 10. Simulation showing robustness of local and population Poisson models to silencing of a subset of neurons.

Left: We created a local Poisson BSN model with 400 neurons, with weights set to perform exact integration, and presented it with a slowly varying 1D stimulus (top). From time t = .4 to t = 1s we artificially silenced 50% of neurons in the negative-weight (blue) population, preventing them from spiking by setting p(spike) = 0 (silencing period indicated by the grey bar). From time t = 1.2 to t = 1.8s we did the same for 50% of neurons in the positive-weight (red) population. Right: Likewise for the population framework.

https://doi.org/10.1371/journal.pcbi.1008261.g010

Related work

Recent literature has explored a variety of other extensions and applications of the BSN framework, including nonlinear dynamical systems and the learning of synaptic weights [48, 49], synaptic plasticity rules [50], and biological extensions like finite timescale synapses [51] and synaptic delays [52]. The BSN framework has also been adapted to other computational problems such as probabilistic computation [53] and sensory adaptation [54].

Our paper is not the first to address the issue of instability in the BSN. Recent work from [52] examined the use of penalties on spiking to reduce ping-ponging (referred to in that paper as “up states”). We found that this strategy required fine-tuning and succeeded in a relatively narrow parameter regime compared to the solutions we proposed here. Other work by by Chalk et al. [36] has argued that oscillations in the brain activity may arise from BSNs with synaptic delays, suggesting that a substantially damped form of ping-ponging may be a signature of efficient computation in neural circuits. This work used the finite timescale dynamics of [51] to implement the BSN with synaptic delays. Similarly to our work, spiking activity in the Chalk model is asynchronous and sparse, and neurons are not prevented from firing synchronously. However, the Chalk model avoids ping-ponging behavior by injecting a carefully tuned amount of noise into membrane potential and using slow membrane time constants (100ms). The added membrane noise degrades the network representation of coding error, forcing a trade-off between stability and accuracy of representation. By contrast, our models do not require tuning membrane potential noise to achieve stability; although firing is stochastic, the membrane potential maintains an accurate representation of the error over a wide range of input and network sizes. Our network also uses membrane time constants with more biophysically plausible ranges (10-20ms).

The topic of balanced networks has also received considerable attention outside the specific BSN framework introduced by [31]. Balanced networks have been proposed as a substrate for working memory [55, 56], probabilistic inference [57, 58], and the control of complex movements [59]. Excitatory-inhibitory balance is also a key topic in the mathematical theory of neural circuit dynamics, where it has been proposed as an explanation for the correlations found in large-scale population activity [60–62]. Finally, a rich literature has focused on the training of spiking neural networks in more general supervised and reinforcement learning settings, where the objective involves task performance or can only be evaluated at the end of a trial [63–67].

Our population Poisson model is similar to work by Eliasmith and Anderson [68] in that the network objective is to minimize the expected squared error between a target function and a network estimate using a spiking neural network. However, their model does so by optimizing for the decoding weights instead of network activity and uses a spiking mechanism that is driven linearly by the magnitude of the input, instead of the error-correcting computational principle used in the BSN. Later work by Eliasmith [69] extended this framework to allow for embedding of the desired dynamics (i.e., the A matrix) into the population activity.

Our work also connects to a rich literature on point process models of neural spike trains. The local Poisson framework draws direct inspiration from the work of [32], which sought to approximate a noisy integrate-and-fire model with an inhomogeneous Poisson process via the so-called “escape-rate approximation”, which refers to the instantaneous probability of noisy membrane potential crossing threshold in a small time window. Subsequent work on the spike response model [39, 70–73] and Poisson generalized linear model [33–35, 74–76] further explored the connection between integrate-and-fire and conditionally Poisson spike train models. The latter are sometimes referred to as “soft-threshold” integrate-and-fire model [77], making the local Poisson model a natural extension of the original BSN model.

Future challenges

Although our proposed frameworks are a step in the direction of biological plausibility, there remain a variety of open challenges. A straightforward way of extending the biological realism of our proposed network is to reformulate it with a separate inhibitory population to comply with Dale’s law, as was done in [36]. Another extension is the development of neurally plausible learning rules and weight patterns. The network we proposed has a static weight matrix with all-to-all connectivity. A more realistic model would allow for sparse connectivity, sign constraints forcing neurons to be purely excitatory or inhibitory, and plausible learning rules that allow weights to change over time as a function of reward signals. There have been successes learning the fast, slow and feed-forward weights through non-local, supervised, control theoretic approaches [48, 49] or, much more recently, through local, Hebbian plasticity rules [78]. Our probabilistic formulation of the Poisson BSN frameworks makes implementing local, Hebbian plasticity rules more tractable, as it opens up the possibility of applying unsupervised learning techniques from traditional machine learning methodology.

We note that our implementation of synaptic delays merely shifts the arrival time of spikes but still involves an instantaneous jump in the filtered spike train dynamics. In future work, we hope to implement time delays with a finite rise time, as was done in [51]. We suspect that since our network is stable with instantaneous jumps in r(t), it will also be stable with smoothly increasing spiking dynamics. Finally, at the level of implementation, we recognize that although our model does not rely on additional spiking conditions or spike-time interpolation, using Poisson firing dynamics instead of the integrate-and-fire approximation represents a slight step away from biological plausibility. We also hope to address the biological plausibility of ‘anti-neurons’ in the population framework.

Another main challenge is the incorporation of nonlinear dynamics. Although the original BSN model was designed to implement linear dynamical systems, it is well known that a wide variety of neural computations are nonlinear. Recent work has proposed an extension of the BSN framework to nonlinear dynamics [30, 48]; combining this approach with conditionally Poisson spiking therefore represents a promising avenue for future work.

Finally, the conditionally Poisson extensions we have proposed provide new opportunities for applying the BSN framework to the interpretation and analysis of real neural data sets. Both the original BSN model and ours assume access to the precise spiking patterns of all the neurons in a population, but real neural recordings typically record only a small fraction of the neurons in a population. Previous work has shown that latent BSN dynamics can be recovered from spike trains in the fully observed case [53]. Other work has discussed the recovery of Poisson generalized linear models from partial recordings [79, 80]. This motivates the development of new methods for identifying balanced network dynamics and computations from partially observed data sets, which may offer fundamental insights into spike-based computation in the brain.

Exponential integrator

Exponential integrators are a class of numerical methods for solving first-order differential equations of the form (28) where −Ay_t is a linear term with a dynamics matrix A, c is a constant, and the nonlinear terms are grouped in g(y_t). We use this method of integration throughout our simulations because it is much more numerically stable than explicit Euler methods. For equations without nonlinear terms, it above can be solved exactly from time 0 to a later time t as (29)

For A = 0, (30)

Using exponential integrators allows us to implement a time delay of magnitude d by having the network spike on the basis of an extrapolated future error at a time t + d. Specifically, we have the network voltage track the error between a predicted x(t + d) and . Eq 16 then becomes (31) where we have assumed that the input to the network, c(t), stays at constant value, c, from t to t + d and that the spike rate evolves passively in the absence of spiking. In the case that A = 0, (32)

In the case of the population framework, the W^⊤ above is replaced by .

Simulation parameters

For our simulations, all parameters are non-dimensionalized. Time is measured in seconds and dt = 0.1ms. All other simulation parameters are shown in Table 1, below.

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Table 1. Simulation parameters.

https://doi.org/10.1371/journal.pcbi.1008261.t001

Fig 2 was generated using the parameter settings described in Boerlin et al, fig. 1C (included below for comparison). The cost terms are μ = 10⁻⁶ and ν = 10⁻⁵ and the voltage decay constant is τ_v = 20. The noise added to the voltage dynamics and the stimulus was Gaussian with σ_v = 10⁻³ and σ_c = 0.01, respectively. When enforcing the constraint that one neuron should spike per time bin, we selected the neuron with the highest voltage above threshold. Another option with similar results would be to select randomly between the ones above threshold.

As a reminder, N is the number of neurons in the network and W is the vector of read-out weights. For the local framework, α is the slope of the exponential nonlinearity, F_max is the saturation, and F_min is the minimum firing rate. For the population framework, κ is the time window over which the network minimizes the error. Finally, A is the dynamics matrix of the linear dynamical system, τ is the decay time constant of the filtered spike trains r, and d is the time delay.

A software implementation of Poisson BSNs, along with code to re-generate figures shown in the manuscript, is available at https://github.com/pillowlab/PoissonBalancedNets.

Cost terms

The original BSN model objective function incorporated two additional cost terms to penalize spiking: a quadratic cost term μ and the linear cost term ν. These terms encouraged the network to use fewer spikes and to distribute spiking more evenly across neurons with large and small output weights. We did not incorporate these in our derivation for clarity, but they are included in our simulations of the BSN.

Including both cost terms into the derivation in the Results section, Eq (4) becomes (33) and the voltage and threshold equations become (34) (35) (36)

The linear cost term (ν) is proportional to the L1 norm of r(t), or . This cost term penalizes the network’s total activity. The quadratic cost term (μ) limits individual neuron firing rates, forcing a spread of activity across all neurons in a population. Over time, the network transfers activity from precise, costly neurons with high firing rates to imprecise, larger weighted neurons to maintain a compromise between efficiency and accuracy of the read-out. Boerlin et al also include a voltage leak term for biological realism.

In our Poisson models, we did not observe the ping-pong effects described in Boerlin et al for the range of parameters we considered, so we don’t need cost terms for network stability. For the local Poisson framework, the cost terms can be included when α and F_max are high enough to cause ping-ponging.

Performance metrics

We use R² as a measure for how well the network read-out is approximating the target variable. The formula for calculating these is (37) for a simulation of time length T, where is the mean of over the entire simulation. The root-mean-squared error (RMSE) is simply (38)

The cross- and auto-correlations between spike trains were calculated as unbiased estimates with a maximum lag of l = 50 time bins according to (39) where x and y are spike trains from two different neurons.