Diffusive Fractional LIF

The Diffusive Fractional Leaky Integrate-and-Fire model provides an alternative numerical approach to fractional-order neuron dynamics. Instead of storing and summing a long voltage history (as in the GL and Caputo methods), it represents the fractional derivative through a set of auxiliary diffusion variables, yielding a system of ordinary differential equations that can be integrated with standard methods.

Diffusive Representation

The core idea behind the diffusive representation is to express the fractional derivative as a weighted integral over an infinite family of exponentially decaying modes. For a function $x(t)$ and fractional order $\alpha \in (0, 1)$:

where $\phi(\omega, t)$ satisfies the first-order ODE:

and the weight function (diffusive kernel) is:

Each mode $\phi(\omega, t)$ is a simple exponential integrator with decay rate $\omega$. The collection of all modes, weighted by $\mu_\alpha(\omega)$, exactly reproduces the fractional derivative. Intuitively, the power-law memory of the fractional derivative is decomposed into a spectrum of exponential memories with different time scales.

Quadrature Approximation

In practice, the continuous integral over $\omega$ is replaced by a finite quadrature with $M$ nodes $\{\omega_j\}_{j=1}^M$ and weights $\{w_j\}_{j=1}^M$:

where each mode evolves as:

The quadrature nodes are typically chosen on a logarithmic grid spanning many decades of frequency, ensuring that both fast and slow dynamics are captured. Common choices include Gauss-Laguerre quadrature or optimized log-spaced grids.

Application to the FLIF

For the fractional LIF equation:

the diffusive representation converts the single fractional-order equation into a system of $M + 1$ first-order ODEs:

with the constraint:

Each $\phi_j$ can be integrated with standard forward Euler, implicit Euler, or any other ODE integrator. The per-step cost is $O(M)$ per neuron, where $M$ is the number of quadrature nodes.

Comparison with GL Approach

The diffusive approach trades exactness for speed: it can approximate the fractional derivative with far fewer auxiliary variables ($M \ll L$) than the GL method requires in history length. For example, $M = 20$ quadrature nodes can approximate the behavior that would require $L = 200$ or more GL history terms, depending on the accuracy requirements and the frequency content of the signal.

Parameters

SPIRES API

cfg.neuron_type = SPIRES_NEURON_FLIF_DIFFUSIVE;

Example Configuration

spires_flif_diffusive_params params = {
    .alpha    = 0.6,
    .num_modes = 20,
    .tau_m    = 20.0,
    .v_rest   = -65.0,
    .v_th     = -50.0,
    .v_reset  = -65.0,
    .bias     = 0.0,
};
 
spires_reservoir_config cfg = {
    .num_neurons       = 500,
    .num_inputs        = 1,
    .num_outputs       = 1,
    .spectral_radius   = 0.9,
    .ei_ratio          = 0.8,
    .input_strength    = 0.1,
    .connectivity      = 0.1,
    .dt                = 1.0,
    .connectivity_type = SPIRES_CONN_RANDOM,
    .neuron_type       = SPIRES_NEURON_FLIF_DIFFUSIVE,
    .neuron_params     = &params,
};

Accuracy Considerations

The accuracy of the diffusive approximation depends on:

1. Number of modes $M$: More modes capture a wider frequency range and reduce quadrature error. Diminishing returns typically set in around $M = 20$—$30$.

2. Quadrature node placement: Log-spaced nodes cover more decades of frequency than linearly spaced nodes. The minimum and maximum frequencies should span the time scales relevant to the problem ($\omega_{\min} \sim 1/T_{\max}$, $\omega_{\max} \sim 1/\Delta t$).

3. Fractional order $\alpha$: For $\alpha$ close to 1, fewer modes are needed because the kernel is more localized. For small $\alpha$, the broad power-law kernel requires more modes for accurate approximation.

The quadrature error is typically $O(M^{-p})$ for some $p > 0$ that depends on the quadrature rule. For optimized log-spaced grids, convergence is rapid, and $M = 20$ often provides errors below 1% relative to the exact fractional derivative.

When to Use Diffusive FLIF

Choose the diffusive FLIF when:

• Speed is critical with fractional dynamics: The $O(M)$ cost with $M \approx 20$ can be significantly cheaper than the $O(L)$ cost with $L \approx 200$ for the GL method.
• Memory is constrained: Each neuron stores $M$ mode amplitudes instead of $L$ voltage history values. For large reservoirs, this difference is significant.
• Approximate memory suffices: If the task does not require exact reproduction of the fractional dynamics and an approximation is acceptable.
• Long simulation times: The diffusive method’s cost does not grow with simulation length (unlike the non-truncated Caputo method), making it efficient for very long time series.

For tasks where exact fractional memory behavior is important or where you need to validate against analytical results, the GL method is preferred.

Reset Behavior

When a spike occurs ($V_n \geq V_{\text{th}}$), the membrane potential is reset to $V_{\text{reset}}$. The diffusion modes $\phi_j$ are also partially reset to maintain consistency. Specifically, the modes are scaled to reflect the discontinuity in $V$ caused by the reset, preventing the accumulation of spurious memory from pre-spike dynamics.

This reset protocol is handled internally by SPIRES and requires no user intervention. It ensures that the diffusive representation remains an accurate approximation of the true fractional dynamics across spike events.

← GL Fractional LIF | Network Topologies: Erdos-Renyi →