Stable recurrent dynamics in heterogeneous neuromorphic computing systems using excitatory and inhibitory plasticity

Abstract

Many neural computations emerge from self-sustained patterns of activity in recurrent neural circuits, which rely on balanced excitation and inhibition. Neuromorphic electronic circuits represent a promising approach for implementing the brain's computational primitives. However, achieving the same robustness of biological networks in neuromorphic systems remains a challenge due to the variability in their analog components. Inspired by real cortical networks, we apply a biologically-plausible cross-homeostatic rule to balance neuromorphic implementations of spiking recurrent networks. We demonstrate how this rule can autonomously tune the network to produce robust, self-sustained dynamics in an inhibition-stabilized regime, even in presence of device mismatch. It can implement multiple, co-existing stable memories, with emergent soft-winner-take-all and reproduce the "paradoxical effect" observed in cortical circuits. In addition to validating neuroscience models on a substrate sharing many similar limitations with biological systems, this enables the automatic configuration of ultra-low power, mixed-signal neuromorphic technologies despite the large chip-to-chip variability.

Fig. 1. On-chip instantiation and validation of an E–I microcircuit on the DYNAP-SE2 neuromorphic platform.

a Image of the DYNAP-SE2 chip, showing the four cores. An E-I network is implemented on chip. Excitatory (Pyr) and Inhibitory (PV) neurons occupy different cores, and are connected by four weight classes, Pyr-to-Pyr (w_ee), Pyr-to-PV (w_ie), PV-to-PV (w_ii), and PV-to-Pyr (w_ei). b Circuit diagram depicting a DPI neuron, with each component within the circuit replicating various sub-threshold characteristics (such as adaptation, leakage, refractory period, etc.) inherent to an adaptive exponential neuron. c Neuron membrane potential traces over time as a response to DC injection. Blue traces show two example cells from the excitatory chip core in response to four current levels. Red traces show two sample PV cells from the inhibitory chip core. The relationship between the measured voltage and the internal current-mode circuit output that represents the membrane potential is given by an exponential function that governs the subthreshold transistor operation mode. In each case, the bottom two traces show neuronal dynamics in the subthreshold range. The top two traces show neuronal dynamics as the membrane potential surpasses the spike threshold (spike added as gray line to indicate when the voltage surpasses threshold). DC injection values are indicated on each trace. d Input-response curves for the resulting Pyr and PV neurons on-chip, averaged over a population of 200 neurons per cell type. The shaded areas encompass  ±1 standard deviations. Inset: zoomed-in view at low firing rates. In (c), it is evident that a higher input current was required to elicit spikes in PV neurons. This phenomenon can be attributed to the higher threshold of PV neurons.

Fig. 2. Cross-homeostasis drives convergence to target firing rates and asynchronous-irregular regime in a E–I network.

a A representative trial of the convergence of excitatory (red) and inhibitory (blue) firing rates during learning. The gray dashed lines show the desired set-point targets. b A representative trial (the same as in a) of the convergence of weight values during learning. The plotted value is nominal, inferred from eq. (1) (see “Methods”). c Raster plot exemplifying on-chip firing activity during a single emulation run at the end of training. Both Pyr and PV neurons successfully converge to an asynchronous-irregular firing pattern at the desired population FRs. d Initial and final firing rates across different chips (n = 57, across two chips) and different initial conditions. For each initial condition, network connectivity was randomized. e Left: relationship between the final w_ee and w_ei values across different iterations. Orange and green colors correspond to two different chips. Right: same for w_ie and w_ii. f Distributions for the 5 ms correlation coefficients between neurons (left) and coefficient of variation (CV²) (right), demonstrating low regularity and low synchrony. The line indicates the mean. The shaded region indicates the standard deviation.

Fig. 3. Neuromorphic emulation of the paradoxical effect confirms inhibition-stabilized regime.

a The paradoxical effect is a well known phenomena in cortical circuits. In the awake resting cortex of mice (upper figure), when inhibitory neurons are optogenetically activated, their firing rates (in red) show a paradoxical decrease in activity during the stimulation, indicative of an inhibition-stabilized network. Adapted from Sanzeni et al.. Similar to the experimental case (bottom figure), numerical simulations of firing rate models with sufficient excitatory gain and balanced by inhibition show the paradoxical effect when the inhibitory population is excited via an external current. b The paradoxical effect can be demonstrated in analog neuromorphic circuits by applying a Poisson input of 250 Hz for 200 ms to the inhibitory units of networks converged to self-sustained activity via cross-homeostasis. The resulting decrease in the firing rate of the inhibitory units demonstrates that the on-chip network is in the inhibition-stabilized regime. No. of trials = 14; the shading around the lines represents the standard deviation in firing rate across trials.

Fig. 4. Ablation of inhibitory learning results in unstable dynamics and impaired training.

a Example network with training results on “fixed” inhibitory synapses. Convergence is impaired if inhibitory plasticity is turned off. Note that during the experiment, the network exhibits “exploding” rates above 300 Hz, corresponding to the neuron’s saturation rates. b Random networks ran with different initialization of all weights (n = 8). The firing rates do not converge to the targets as precisely as in Fig. 2a, having a much higher root mean square error from the target (RMS on $Ē$: 6.39, on $Ī$: 35.11).

Fig. 5. Results obtained using Mackwood et al.’s rule.

a Convergence of the firing rate to stable self-sustained dynamics. Note that, in this case, there is no inhibitory target. b Convergence of the weight values. w_ii and w_ee are fixed and thus do not show any dynamics under this rule. c Initial and final firing rate values over many trial runs. $Ē$ mostly converges to the target, while, unlike with cross-homeostasis, the final $Ī$ is left to vary freely (n = 34).

Fig. 6. Simultaneous training of multiple clusters.

a Schematic of clusters on the chip. One core contains excitatory populations, and another contains inhibitory populations. We implemented five E/I networks, with each network comprising 50 excitatory and 15 inhibitory neurons. As per table 1, the connection probability within these networks is 0.35. To account for the network’s scaling, we defined higher set-points. Note that these clusters do not interconnect with one another. b Rate convergence for five clusters sharing the same nominal weights. Each color represents one cluster.

Fig. 7. Training result with a memory implanted in the network.

a Random recurrent E/I network with a neural ensemble (size = 32, green neurons). b Convergence of both excitatory (Pyr) and inhibitory (PV) populations to their respective set-points. A subset of the weights is manually changed to instantiate a new ensemble at iteration 250, well after the network has converged to its set-point. A jump in activity is observed in both Pyr and PV cells due to the presence of the newly implanted memory. c Schematic illustration of the process of re-balancing a weight matrix after implanting a new memory, instantiated as a network of six neurons containing an ensemble of three neurons. This is only indicative, as in the current chip it is not possible to read the weight matrix for visualization. d The nominal weight value for 32 neurons (w_ee) is increased by 0.0088 μA at the time point highlighted with the dotted line. All weights undergo alteration again to compensate for the presence of the new ensemble, bringing the network back to the set-point after the memory is implanted. However, the memory remains intact, as evidenced by higher recurrent weight (w_memory) compared to the baseline excitatory connectivity (w_ee).

Fig. 8. Winner-take-all dynamics with three sub-clusters implanted in the network.

a Random recurrent E/I network with three neural ensembles (size = 50, green, orange and purple neurons). The stimulation protocol (top left) shows how different sub-clusters receive inputs over the course of emulation. b Firing rates of all clusters when each cluster is stimulated with either high (300 Hz) or low frequency (60 Hz) inputs. The cluster receiving the high-frequency input dominates the competition. We demonstrate that no special tuning is necessary to achieve this winner-takes-all (WTA) phenomenon. c Raster plot illustrating spike activity for all neurons in different clusters. d Activity in an extended version of the memory network with 10 sub-clusters, each with 20 neurons. Here every cluster is stimulated independently, showing that up to 10 memories can be individually recalled.