Josephson junction simulation of neurons

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.

Figure 1

Circuit diagram for the JJ Neuron (left loop) connected to a model chemical synapse (right loop). In general, many synapses could connect to a single JJ Neuron. Orientation of junction phases is chosen for clockwise current in the left loop.

Figure 2

(Color online) (a) Time profile for action potentials in the JJ Neuron model. Parameter values for JJ Neuron calculations are $\lambda =0.1$, $\Gamma =1.5$, ${\Lambda }_s={\Lambda }_b=0.5$, $\eta =1$. Input dc current is $i_{in}=0.21$. (b) Time profile for action potentials in the Hodgkin-Huxley model. The flux (black solid) in the JJ Neuron corresponds to the membrane potential (black solid) in the Hodgkin-Huxley model. Similarly, the voltages $v_p$ (red dashed) and $-v_c$ (blue dot-dashed) in the JJ Neuron model correspond to the currents $-I_{\text{Na}}$ (red dashed) and $I_{\text{K}}$ (blue dot-dashed) in Hodgkin-Huxley model. Parameter values for all Hodgkin-Huxley calculations are $C_m=1.01\mu \text{F}/{\text{cm}}^2$; ${\overline{G}}_{\text{Na}}=120\text{mS}/{\text{cm}}^2$; ${\overline{G}}_{\text{K}}=36\text{mS}/{\text{cm}}^2$; $G_L=0.3\text{mS}/{\text{cm}}^2$; $E_{\text{Na}}=50\text{mV}$; $E_{\text{K}}=-77\text{mV}$; $E_L=-54.4\text{mV}$; and $T=18.5°\text{C}$ unless stated otherwise. In this simulation an external stimulating current of 236 nA is applied. The Hodgkin-Huxley neurons are modeled as isopotential cylinders with lengths and diameters of $500\mu \text{m}$.

Figure 3

(Color online) The threshold response of flux time-traces as the strength of impulse input signals increases. Stronger inputs (initially higher curves) lead to action potentials. The inset graph shows how peak response depends on the strength of the input pulse. Parameter values as for Fig. 2a except $\Gamma =1.0$. Input current is a single square pulse of width 5 and varied height initiated at $t=30$.

Figure 4

(Color online) Frequency response to increased input strength. The input current (red linear) and flux response (black oscillating) curves show the frequency response due to a slow linear increase in input strength. Parameters values are as in Fig. 3 except for $\Gamma$. The top plot $\left(\Gamma =0.9\right)$ shows frequency largely independent of input strength after onset of repetitive spiking. The bottom plot $\left(\Gamma =1.5\right)$ shows a low frequency at onset which increases as the input strength in increased. This demonstrates that the JJ Neuron can model either class 1 or class 2 neurons depending on parameter values.

Figure 5

(Color online) (a) Response to twin pulse inputs with delay shows the refractory period. On the left, the time of the first (lower blue) and second (upper red) response peak is graphed against the delay between input pulses. The vertical dashed line represents the delay below which no second pulse is created. On the right, time profiles are shown for input currents (green dashed) and output voltages (blue solid). In (b) two pulses are generated while in (c), the delay is too small and no second pulse arises. Parameter values as for Fig. 2a except $\Gamma =1.0$. Input pulses are twin square pulses of height 0.54, width 5 and varied delay between pulses. The first pulse initiates at $t=30$.

Figure 6

(Color online) Excitatory synaptic coupling. One JJ neuron in blue (solid) drives another in black (dashed). The system is initially quiet. At time $t=20$, a constant input current (red lower) causes the first neuron to fire repetitively. Excitatory coupling induces repetitive firing of the second neuron. Parameter values as for Fig. 2a except $i_b=2.0$, $\Gamma =2.0$. Synaptic parameters are $\Omega =1.0$, $Q=0.05$, ${\Lambda }_{syn}=0.3$, $r_{12}=1.4$. Input current to first neuron is a dc current of $i_{in}=0.3$ initiated at $t=20$.

Figure 7

(Color online) Inhibitory synaptic coupling. The second neuron (upper dashed) is configured to fire repetitively. When the first neuron (middle blue) is activated by an external input current (lower red), it inhibits the second. Both return to their original state after the external stimulus to the first neuron is removed. Parameter values as for Fig. 6 except $i_{b1}=-1.9$, $i_{b2}=1.76$, ${\Lambda }_{s2}=0.65$, and ${\lambda }_{p2}=0.35$. This pushes the second neuron into a repetitive firing mode. Synaptic parameters are also the same except $r_{12}=0.6$. Input current to first neuron is a single square pulse of height 0.3, width 500, at $t=175$. For illustrative purposes, the first neuron’s flux is shifted down by $-0.7$ in the plot.

Figure 8

(Color online) Behavior of two Hodgkin-Huxley model neurons coupled by a simple excitatory synapse model. The synapse uses a positive “alpha” function [16] with maximum current 400 nA and time scale 0.1 ms which is triggered at $-30\text{mV}$ on the downward slope of the presynaptic neuron’s action potential. The presynaptic neuron in blue (solid oscillating) is made to fire repetitively by means of a constant current input (lower red curve) initiated at time 5 ms. Excitatory coupling then causes repetitive firing of the postsynaptic neuron (dashed).

Figure 9

(Color online) Behavior of two Hodgkin-Huxley neurons coupled with an inhibitory synapse model using a negative alpha function with maximum current 215 nA and time scale 1 ms. The second neuron (dashed) has $E_L$ set to $-15\text{mV}$ to make it fire repetitively in the absence of external inputs, while the first has $E_L$ set at the standard $-54.4\text{mV}$ value to make it normally quiescent. When the first neuron (middle blue) fires as the result of an external input current (lower red), it inhibits the second. Both return to their original state after the external stimulus to the first neuron is removed. For illustrative purposes, the second neuron’s membrane potential is shifted upwards by 110 mV in the plot.