Neuron Model in Axon

This document details the spiking neuron model used in Axon, which implements the STICK (Spike Time Interval Computational Kernel) computational paradigm. It emphasizes temporal coding, precise spike timing, and synaptic diversity for symbolic computation.

1. 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

2. Neuron Dynamics

Each neuron maintains four internal state variables:

The membrane potential evolves according to:

\tau_m \frac{dV}{dt} = g_e + \text{gate} \cdot g_f

where:

• τm is the membrane time constant
• g_e is the persistent excitatory input
• g_f is the fast decaying input, gated by gate The neuron spikes when V exceeds a threshold Vt, at which point it emits a spike and resets its state. After a spike, the neuron resets:

V → Vreset ge → 0 gf → 0 gate → 0

This reset guarantees clean integration for subsequent intervals.

4. Synapse Types

Axon supports four biologically inspired synapse types:

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

5. Implementation Summary

Class: AbstractNeuron

• Implements update logic for ge, gf, and gate
• Defines update_and_spike(dt) for simulation cycles
• Supports receive_synaptic_event(type, weight)

Class: ExplicitNeuron

• Inherits from AbstractNeuron
• Tracks:
  • spike_times[]
  • out_synapses[]
• Implements reset() after spike emission

Class: Synapse

• Defines:
  • pre_neuron, post_neuron
  • weight, delay, type
• Used to construct event-driven spike queues with delay accuracy

6. Temporal Coding & Integration

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

Integration periods in neurons align with computation windows:

• ge: accumulates static value during inter-spike interval
• gf + gate: used for exponential/logarithmic timing
• V: compares integrated potential to threshold for spike emission

These dynamics enable symbolic operations such as memory, arithmetic, and differential equation solving.

7. Numerical Parameters

Typical parameter values used in Axon:

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

8. Benefits of This Model

• 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 like ADA

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. The neuron emits a spike when its membrane potential V(t) reaches the threshold Vt = 10.0 mV, after which it resets.

Synapse Events Timeline

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.

Summary of Synapse Effects

Spike Dynamics

When V ≥ Vt, the neuron:

• Spikes
• Logs spike time
• Resets all internal state to baseline

You can see these spikes as red dots at the threshold line in the animation.

References

• Lagorce & Benosman (2015): Spike Time Interval Computational Kernel
• Axon SDK Source:
  • Neuron model: axon/elements.py
  • Event logic: axon/events.py
  • Simulator integration: axon/simulator.py