Neuron Model in Axon

This document details the spiking neuron model used in Axon, which implements the STICK computational paradigm. STICK uses temporal coding, precise spike timing, and synaptic diversity for symbolic and deterministic computation.

STICK: Spike Time Interval Computational Kernel, A Framework for General Purpose Computation using Neurons, Precise Timing, Delays, and Synchrony.

Overview

Axon simulates event-driven, integrate-and-fire neurons with:

• Millisecond-precision spike timing
• Multiple synapse types with distinct temporal effects
• Explicit gating to modulate temporal dynamics

The base classes are:

• AbstractNeuron: defines core membrane equations
• ExplicitNeuron: tracks spike times and enables connectivity
• Synapse: defines delayed, typed connections between neurons

Neuron Dynamics

Each neuron maintains five internal state variables:

The membrane potential evolves following the differential equation:

\[ \tau_m \frac{dV}{dt} = g_e + \text{gate} \cdot g_f \] \[ \frac{dg_e}{dt} = 0 \] \[ \tau_f \frac{dg_f}{dt} = -g_f \]

When the membrane potential surpasess a threshold, V > Vt, the neuron emits a spike and resets:

V → Vreset
ge → 0
gf → 0
gate → 0

Reset guarantees clean operation for subsequent intervals.

Synapse Types

The neuron model supports four synapse types with a certain weight (w).

Each synapse also includes a configurable delay, enabling precise temporal computation.

Numerical Parameters

Typical neuron parameter values used in Axon:

Units are in milliseconds and millivolts, matching real-time symbolic processing and neuromorphic feasibility.

Benefits of This Model

This neuron model is designed for interval-coded values. Time intervals between spikes directly encode numeric values.

The neuron model has dynamic behaviours that eenable symbolic operations such as memory, arithmetic, and differential equation solving. The dynamics of this neuron model forms a Turing-complete computation framework (for in depth information, refer to the STICK paper).

This neuron model has the following characteristics:

• Compact: Minimal neurons required for functional blocks
• Precise: Accurate sub-millisecond spike-based encoding
• Composable: Modular design supports hierarchical circuits
• Hardware-Compatible: Ported to digital integrate-and-fire cores

Neuron Model Animation

This animation demonstrates how a single STICK neuron responds over time to different synaptic inputs. Each input type (V, ge, gf, gate) produces distinct changes in membrane dynamics. Synapse-type ge produces a linear increase of V and gf an exponential one.

Summary of Synapse Effects

Event-by-event explanation

t = 20 ms - V(10.0)

• A V-synapse adds +10.0 mV to V instantly.
• Since Vt = 10.0, this causes immediate spike.
• The neuron resets: V → 0, ge, gf, gate → 0.

Effect: Demonstrates a direct spike trigger via instantaneous voltage jump.

t = 60 ms — ge(2.0)

• A ge-synapse applies constant input current.
• Voltage rises linearly over time.
• Alone, this isn’t sufficient to reach Vt, so no spike occurs yet.

Effect: Shows the smooth effect of continuous integration from ge-type input.

t = 100 ms — gf(2.5) and gate(1.0)

• A gf-synapse delivers fast-decaying input current.
• A gate-synapse opens the gate (gate = 1), activating gf dynamics.
• Voltage rises nonlinearly as gf initially dominates, then decays.
• Combined effect from earlier ge and gf causes a spike shortly after.

Effect: Demonstrates exponential integration (gf) gated for a temporary burst.

t = 160 ms — V(2.0)

• A small V-synapse bump of +2.0 mV occurs.
• This is not enough to cause a spike, but it shifts V upward instantly.

Effect: Shows subthreshold perturbation from a V-type synapse.

t = 200 ms — gate(-1.0)

• The gate is closed (gate = 0), disabling gf decay term.
• Any remaining gf is no longer integrated into V.

Effect: Demonstrates control logic: gf is disabled, computation halts.