With the goal of understanding the intricate behavior and dynamics of collections of neurons, we present superconducting circuits containing Josephson junctions that model biologically realistic neurons. These "Josephson junction neurons" reproduce many characteristic behaviors of biological neurons such as action potentials, refractory periods, and firing thresholds. They can be coupled together in ways that mimic electrical and chemical synapses. Using existing fabrication technologies, large interconnected networks of Josephson junction neurons would operate fully in parallel. They would be orders of magnitude faster than both traditional computer simulations and biological neural networks. Josephson junction neurons provide a new tool for exploring long-term large-scale dynamics for networks of neurons.

. The action potential of the unmyelinated nerve is metabolically expensive. Using the energetic cost per unit length for the biophysically modeled action potential of the squid giant axon, we analyze this cost and identify one possible optimization. The energetic cost arising from an action potential is divided into three separate components: 1) the depolarization of the rising phase; 2) the hyperpolarization of the falling phase; and 3) the largest component, the overlapping of positive and negative currents, which has no electrical effect. Using both the Hodgkin-Huxley (HH) model and an improved version of the HH model (HHSFL), we investigate the variation of these three components as a function of easily evolvable parameters, axon diameter and ion channel densities. Assuming conduction velocity is well designed for each organism, the energy component associated with the rising phase attains a minimum near the biological values of the diameter and channel densities. This optimization is explained by the membrane capacitance per unit length. The functional capacitance is the sum of the intrinsic membrane capacitance and the gating capacitance associated with the sodium channel, and this capacitance minimizes at nearly the same values of diameter and channel density. Because capacitance is temperature independent and because this result is independent of the assumed velocity, the result generalizes to unmyelinated mammalian axons. That is, channel density is arguably an evolved property that goes hand-in-hand with the evolutionary stability of the sodium channel. I N T R O D U C T I O NIn the nervous system, the action potential is used for long-distance information transmission. Delivery of such information in a timely fashion requires an action potential of sufficient velocity. On the other hand, sufficient velocity has its costs. In what follows, we assume that across species and across the life span of the organism the velocity of any axon is appropriate to its role in information processing.In the neuropil of neocortex, where axons must be unmyelinated if each one is to make several thousand sequential or neighboring synapses, the metabolic costs are surprisingly large. Attwell and Laughlin (2001) estimated that 75% of the adenosine triphosphate (ATP) consumed by neurons in the rat brain is used for communication and computation. Of this, half is used by the unmyelinated axons.This metabolic perspective contrasts with and, as we will see, ultimately complements Hodgkin's conjectured constraint on action potential velocity. Both Hodgkin (1975) and Adrian (1975) proposed that the gating charge movement that inevitably accompanies rapid activation of a voltage-dependent channel leads to an optimal density of fast Na ϩ channels. This optimization occurs because the movement of charge specifically restricted to the transmembrane voltage field contributes, albeit transiently, to membrane capacitance. Because increasing capacitance slows action potential propagation, Hodgkin proposed that the Na ϩ channel dens...

Conventional digital computation is rapidly approaching physical limits for speed and energy dissipation. Here we fabricate and test a simple neuromorphic circuit that models neuronal somas, axons, and synapses with superconducting Josephson junctions. The circuit models two mutually coupled excitatory neurons. In some regions of parameter space the neurons are desynchronized. In others, the Josephson neurons synchronize in one of two states, in-phase or antiphase. An experimental alteration of the delay and strength of the connecting synapses can toggle the system back and forth in a phase-flip bifurcation. Firing synchronization states are calculated >70 000 times faster than conventional digital approaches. With their speed and low energy dissipation (10^{-17}J/spike), this set of proof-of-concept experiments establishes Josephson junction neurons as a viable approach for improvements in neuronal computation as well as applications in neuromorphic computing.

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