Caputo Fractional LIF

The Caputo Fractional Leaky Integrate-and-Fire (FLIF) model replaces the integer-order time derivative in the classical LIF equation with a fractional derivative of order $\alpha \in (0, 1]$. This introduces power-law memory into the neuron dynamics, allowing the membrane potential to retain information about past inputs over much longer time scales than the exponential decay of integer-order models.

Fractional Derivative: The Caputo Definition

The Caputo fractional derivative of order $\alpha$ for a function $x(t)$ is defined as:

where $\Gamma(\cdot)$ is the Euler gamma function and $\dot{x}(\tau)$ denotes the ordinary first derivative.

Key properties of the Caputo derivative:

• When $\alpha = 1$, it reduces to the ordinary first derivative: ${}_C D_t^1 x(t) = \dot{x}(t)$.
• For $\alpha \lt 1$, the integral kernel $(t - \tau)^{-\alpha}$ is singular at $\tau = t$ and decays as a power law, weighting recent history more heavily than distant history — but with a much slower decay than an exponential.
• Initial conditions are specified in terms of integer-order derivatives (e.g., $x(0)$), which is physically meaningful and distinguishes the Caputo derivative from the Riemann-Liouville definition.

The Fractional LIF Equation

The FLIF equation replaces the integer-order derivative in the LIF model:

This is identical in form to the classical LIF, but the left-hand side is now a fractional derivative. The physical interpretation is that the neuron’s membrane exhibits anomalous diffusion rather than normal diffusion:

• In a standard capacitor (integer-order), charge redistribution follows exponential dynamics.
• In a fractional capacitor (fractional-order), charge redistribution follows power-law dynamics, which models non-ideal dielectric materials, heterogeneous ion channel populations, or complex dendritic morphologies.

Memory and the Fractional Order

The fractional order $\alpha$ directly controls the memory characteristics of the neuron.

The impulse response of the fractional LIF (without threshold) is governed by the Mittag-Leffler function:

The Mittag-Leffler function $E_\alpha(z)$ interpolates between a stretched exponential for small $t$ and a power-law tail for large $t$:

This means that even long after a stimulus, the membrane potential retains a non-negligible trace of the input. The rate of this power-law decay is controlled by $\alpha$:

• $\alpha \to 0^+$: Extremely slow decay. The neuron has a very long memory but is sluggish in responding to new inputs.
• $\alpha = 0.5$: Intermediate regime with algebraic $t^{-1/2}$ decay envelope.
• $\alpha \to 1^-$: Exponential decay is recovered, and the neuron behaves like a classical LIF.

Numerical Discretization

The Caputo FLIF in SPIRES uses a discretization based on the L1 scheme, which approximates the Caputo integral using a quadrature over the history of the function. At time step $n$ with step size $\Delta t$:

This requires storing the complete voltage history up to a truncation length $L$, making the per-step cost $O(L)$ in both time and memory. The Caputo formulation ensures that the initial condition $V(0) = V_{\text{rest}}$ is directly applied.

Parameters

SPIRES API

cfg.neuron_type = SPIRES_NEURON_FLIF_CAPUTO;

Example Configuration

spires_flif_params params = {
    .alpha   = 0.7,
    .tau_m   = 20.0,
    .v_rest  = -65.0,
    .v_th    = -50.0,
    .v_reset = -65.0,
    .bias    = 0.0,
    .L       = 200,
};
 
spires_reservoir_config cfg = {
    .num_neurons       = 300,
    .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_SMALL_WORLD,
    .neuron_type       = SPIRES_NEURON_FLIF_CAPUTO,
    .neuron_params     = &params,
};

Caputo vs. Riemann-Liouville

SPIRES uses the Caputo definition rather than the Riemann-Liouville (RL) definition for several reasons:

1. Physical initial conditions: The Caputo derivative allows initial conditions to be specified as $V(0) = V_{\text{rest}}$, which is directly measurable. The RL definition requires fractional-order initial conditions that have no clear physical interpretation.

2. Constant function behavior: The Caputo derivative of a constant is zero, as expected physically. The RL derivative of a constant is not zero for $\alpha \lt 1$.

3. Neuroscience convention: The Caputo formulation is standard in the fractional-order neuron modeling literature.

Computational Cost

The Caputo FLIF has cost $O(L)$ per neuron per time step, where $L$ is the history length. For a reservoir of $N$ neurons:

• Memory: $O(N \cdot L)$ to store voltage histories
• Time per step: $O(N \cdot L)$ for the quadrature sums, plus $O(N^2)$ or $O(N \cdot \text{nnz})$ for synaptic current computation

For very long histories, consider the GL discretization, which uses the same $O(L)$ cost per step but with precomputed binomial coefficients that may be slightly more cache-friendly, or the Diffusive representation, which trades exact memory for an $O(M)$ approximation with $M \ll L$ quadrature nodes.

When to Use Caputo FLIF

Choose the Caputo FLIF when:

• Theoretical clarity is important: The Caputo formulation has the most intuitive mathematical properties and the largest body of analytical results.
• You need to compare with published work: Most fractional neuron modeling papers use the Caputo definition.
• Moderate history lengths suffice: If $L$ does not need to be extremely large (e.g., $L \lt 500$), the direct Caputo discretization is straightforward and accurate.

For production use with large reservoirs and long histories, the GL formulation is generally preferred due to its efficient coefficient computation and circular buffer implementation.

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