SOEN Circuit Physics and Modeling

Dendritic Circuit Structure

Each SOEN dendrite consists of two fundamental components connected by a superconducting quantum interference device (SQUID):

Receiving Loop (R)

• Couples flux $\varphi$ from upstream dendrites or neurons
• Flux coupling enables inter-dendritic communication
• Input drives the dendritic SQUID circuit

Integration Loop (I)

• Contains inductance $L$, resistance $r$, and optionally capacitance $C$
• Stores dimensionless signal $s = I/I_c$ (current normalized by critical current)
• Provides temporal integration with time constant $\tau = \beta L/r$

Flux Coupling Between Dendrites

Dendrites communicate through flux coupling via mutual inductance. The coupling flux to dendrite $i$ from all other dendrites $j$ is:

$$\varphi_i = \sum_j J_{ij} s_j + \varphi_i^{\text{ext}}$$

Where:

• $J_{ij}$: Static coupling matrix determined by mutual inductance from transformers on collection coils
• $s_j$: Dimensionless signal (current) in integration loop of dendrite $j$
• $\varphi_i^{\text{ext}}$: External flux drive to dendrite $i$

Phenomenological Model Equations

The temporal dynamics of each dendrite follow a unified differential equation:

$$\frac{ds}{dt} = \gamma , g(\varphi,s) - \frac{s}{\tau}$$

Where:

• $s$: Dimensionless signal in integration loop
• $\varphi$: Total flux coupled into receiving loop
• $\gamma = 1/\beta$: Dimensionless gain parameter ($\beta = 2\pi LI_c/\Phi_0$)
• $\tau = \beta/\alpha$: Time constant ($\alpha = r/r_{jj}$, resistance ratio)
• $g(\varphi,s)$: Source function encoding circuit physics

Source Functions $g(\varphi,s)$

The source function captures the rate of flux-quantum production by the SQUID and depends on the circuit type:

Dendritic Source Function $g_d(\varphi,s)$

• For dendrites receiving flux from other dendrites
• Monotonically increasing with input flux $\varphi$
• Monotonically decreasing with stored signal $s$ (saturation effect)
• Parameters: dendritic bias current $i_d$

Neuronal Source Function $g_n(\varphi,s)$

• For dendrites receiving spikes from upstream neurons
• Encodes soma threshold crossing, transmitter dynamics, and synaptic response
• Parameters: neuronal bias current $i_n$, dendritic bias current $i_d$
• Somatic threshold $s_{th}$ (typically 0.2)

Circuit Parameters and Physical Scales

Dimensionless Quantities

• $\beta = 2\pi LI_c/\Phi_0$: Loop inductance parameter ($10^2$ to $10^5$)
• $\alpha = r/r_{jj}$: Resistance ratio
• $\gamma = 1/\beta$: Gain parameter
• $\tau = \beta/\alpha$: Integration time constant

Physical Time Scales

• Josephson dynamics: ~1-10ps (ignored in phenomenological model)
• Dendritic integration: 50ns to 6.25μs ($\tau$ parameter)
• Inter-spike intervals: 50ns minimum (20MHz maximum firing rate)
• Synaptic response: 1-10ns (abstracted in neuronal source function)

Bias Current Effects

Bias currents ($i_b$, $i_n$, $i_d$) are circuit parameters that modify source function shapes:

• Dendritic bias $i_d$: Affects flux threshold and signal saturation in $g_d(\varphi,s)$
• Neuronal bias $i_n$: Shifts flux threshold for spike generation
• Synaptic bias: Influences downstream dendritic response characteristics

Lagrangian Foundation

The phenomenological model derives from circuit Lagrangian principles:

$$\mathcal{L} = \mathcal{T} - \mathcal{V} = \frac{1}{2}L\dot{q}^2 - \frac{1}{2}\frac{q^2}{C}$$

With damping $\mathcal{F} = \frac{1}{2}r\dot{q}^2$ and electromotive force $\varepsilon = \Phi_0 G_{fq}$ representing flux-quantum production rate. This provides a rigorous physical foundation based on least-action principles.

Comparison to Classical Neurodynamics

The SOEN model maps directly onto classical neurodynamic equations:

$$C_i \frac{du_i}{dt} = \sum_j T_{ij} f_j(u_j) - \frac{u_i}{R_i} + I_i$$

With correspondence:

• Membrane potential $u \leftrightarrow$ Flux $\varphi$
• Transfer function $f(u) \leftrightarrow$ Source function $g(\varphi,s)$
• Synaptic coupling $T \leftrightarrow$ Mutual inductance $J$

Single-Photon Communication

SOENs communicate using binary optical signals at the fundamental quantum limit - single photons per synaptic event. This approach minimizes optical power requirements while maximizing communication efficiency.

Superconducting-Nanowire Single-Photon Detectors (SPDs)

Each synapse in a SOEN utilizes an SPD - a current-biased strip of superconducting wire that detects individual near-infrared photons. SPDs provide:

• Quantum efficiency: >90% (93% demonstrated) for detecting single photons
• Ultra-low noise: Negligible dark count rates at 4K
• High speed: Response times <1ns
• No static power: Dissipationless superconducting operation

Silicon Light Sources at 4K

Unlike room-temperature systems, SOEN operation at 4K enables silicon LEDs using defect-based dipole emitters:

• Waveguide-integrated: Directly coupled to on-chip photonic routing
• Low power: ~1fJ per optical pulse (single photons)
• CMOS compatible: Fabricated with standard silicon processes
• Cryogenic efficiency: Enhanced performance at liquid helium temperatures

Waveguide-Integrated Photonics

Multi-Planar Routing

SOENs employ multiple planes of dielectric waveguides for dense optical routing:

• Waveguide pitch: 1.5μm spacing between parallel guides
• Multiple planes: 6+ layers of waveguides per wafer
• Low loss: <0.1dB/cm propagation loss in silicon nitride
• Direct fan-out: No shared switching infrastructure needed

Scaling Architecture

• Wafer-scale: ~1 million neurons per 300mm wafer
• Power consumption: ~10MW for brain-scale system (including cooling)
• Vertical stacking: Free-space optical links between wafer layers
• Hierarchical organization: Columnar structure mimicking cerebral cortex