Relating Superconducting Optoelectronic Networks to Classical Neurodynamics

Abstract

The circuits comprising superconducting optoelectronic synapses, dendrites, and neurons are described by numerically cumbersome and formally opaque coupled differential equations. Reference 1 showed that a phenomenological model of superconducting loop neurons eliminates the need to solve the Josephson circuit equations that describe synapses and dendrites. The initial goal of the model was to decrease the time required for simulations, yet an additional benefit of the model was increased transparency of the underlying neural circuit operations and conceptual clarity regarding the connection of loop neurons to other physical systems. Whereas the original model simplified the treatment of the Josephson-junction dynamics, essentially by only considering low-pass versions of the dendritic outputs, the model resorted to an awkward treatment of spikes generated by semiconductor transmitter circuits that required explicitly checking for threshold crossings and distinct treatment of time steps wherein somatic threshold is reached. Here we extend that model to simplify the treatment of spikes coming from somas, again making use of the fact that in neural systems the downstream recipients of spike events almost always perform low-pass filtering. We provide comparisons between the first and second phenomenological models, quantifying the accuracy of the additional approximations. We identify regions of circuit parameter space in which the extended model works well and regions where it works poorly. For some circuit parameters it is possible to represent the downstream dendritic response to a single spike as well as coincidences or sequences of spikes, indicating the model is not simply a reduction to rate coding. The governing equations are shown to be nearly identical to those ubiquitous in the neuroscience literature for modeling leaky-integrator dendrites and neurons.

Figures (12)

• Figure 1: Circuits considered in the phenomenological model. (a) Two dendritic integration loops (I) connected to a collection coil (labeled C), which couples flux ($\phi$) into the receiving loop of another dendrite (R). The dendrite itself is the combination of the R and I loops. (b) The dendritic SQUID and output $LRC$ loop with circuit parameters labeled. The objective of the phenomenological model is to represent the activity of this circuit without reference to the rapid JJ dynamics. (c) The circuit considered in the model, where the activity of the SQUID has been abstracted into the voltage source $\bar{V}$.
• Figure 2: Source functions for direct connection from soma to dendrite. (a,b) Dendrite source functions for two different bias currents. (c-f) Neuron source functions for two different neuronal and two different dendritic bias currents.
• Figure 3: Comparison of circuit response as calculated by first-principles circuit equations and phenomenological model. (a) The flux applied to the dendritic receiving loop. (b) The signal in the dendritic integration loop as calculated with the circuit equations (black, solid trace) and the first phenomenological model (green, dashed trace). (c) Zoom of a small region from (b) showing the the difference between the two models. For these calculations, $\beta_\mathrm{d}/2\pi = 10^3$, $\tau_\mathrm{d} = 250$ ns, $\chi^2 = 2.5\times 10^{-6}$, $i_\mathrm{d} = 1.7$.
• Figure 4: Schematic of the phenomenological extension. (a) The original phenomenological model, with dendrites ($\mathsf{D}$) coupled into a collection coil ($\mathsf{C}$), which feeds into the neuron cell body ($\mathsf{N}$). The input flux to dendrites is $\phi_\mathrm{d}$, and that to the neuron is $\phi_\mathrm{n}$. Upon reaching somatic threshold, the neuron activates the refractory dendrite ($\mathsf{D_r}$), which delivers inhibitory feedback to the soma to temporarily increase the threshold for firing, and the transmitter ($\mathsf{T}$) is activated, which produces a photonic action potential that delivers light to downstream synapses ($\mathsf{S}$) that are coupled to output dendrites with time constant $\tau_\mathrm{d}$ that can vary across dendrites. In response to sustained input flux, the neuron will produce a train of action potentials, as illustrated by the stream of spikes between the transmitter and downstream synapses. (b) The full phenomenological model. In contrast to (a), the output flux from the collection coil, $\phi_\mathrm{n}$, is given directly to downstream dendrites, and these dendrites produce signals based on this flux using a different source function, and hence these boxes are colored red instead of yellow as used for dendrites not receiving flux directly from neurons. For these dendrites, $\phi_\mathrm{d} = \phi_\mathrm{n}$.
• Figure 5: Second phenomenological model concept illustrated with the response to an input flux step function. (a) The soma dynamics as calculated with the model of Ref. shainline2023phenomenological. The externally applied flux input to the soma is a step function above the somatic flux threshold. The total flux is the external input plus the feedback flux from the refractory dendrite. (b) The signal generated in the neuronal integration loop due to the flux drive. The saw-tooth behavior results from purging of signal upon reaching threshold, followed by refraction and reaccumulation. (c) Comparison of the responses of downstream recipient dendrites calculated with the spiking and phenomenological models. $s_\mathrm{out}$ is the signal integrated in the output dendrite. The insets show the neural circuit schematics for the two cases. The spiking model requires calculation of the soma traces shown in (a) and (b) as well as the response of the synaptic single-photon detector, while the phenomenological model only uses the input flux signal, with the dynamics of the neuronal integration loop, refractory dendrite, and synaptic detector encoded in $g_\mathrm{n}$. The $\chi^2$ between the signals in the output dendrites calculated with the spiking and phenomenological models was $1.91\times 10^{-3}$. $\beta_\mathrm{in}/2\pi = 10^3$, $\tau_\mathrm{in} = 250$ ns, $\beta_\mathrm{out}/2\pi = 10^4$, $\tau_\mathrm{out} = 1.25$ µ s. For the input dendrite, soma, and output dendrite, $i_\mathrm{b} = 1.7$.