LIF Dynamics

The Leaky Integrate-and-Fire (LIF) neuron is the workhorse model of computational neuroscience. It captures the essential dynamics of a biological neuron — passive membrane decay, synaptic integration, and threshold-triggered spiking — while remaining analytically tractable and computationally efficient. This page develops the classical LIF model, extends it to fractional order, derives the Grunwald-Letnikov discretization used in SPIRES, and analyzes the effects of the fractional order $\alpha$ on neural dynamics.

The Classical LIF Neuron

Biophysical Form

A biological neuron’s membrane can be modeled as a parallel RC circuit. The membrane capacitance $C$ charges in response to input current $I(t)$, while the membrane resistance $R$ causes passive leakage toward the resting potential $V_{\text{rest}}$. The resulting equation is:

When the membrane potential $V(t)$ reaches the threshold $V_{\text{th}}$, the neuron emits a spike and $V$ is reset to $V_{\text{reset}}$. The membrane time constant is $\tau_m = RC$.

Normalized Form

Dividing Equation (1) by $C$ and using $\tau_m = RC$:

where $b$ is an optional bias current. This is the standard form used in most reservoir computing implementations. The dynamics are straightforward:

• Leak: The term $-(V - V_{\text{rest}})$ drives $V$ toward the resting potential with time constant $\tau_m$.
• Integration: The input current $I(t)$ charges the membrane.
• Spike-and-reset: When $V \geq V_{\text{th}}$, emit a spike and set $V \leftarrow V_{\text{reset}}$.

The solution in the absence of spiking and for constant input $I$ is an exponential approach to equilibrium:

The exponential decay $e^{-t/\tau_m}$ means the classical LIF neuron has a single characteristic timescale. Memory of past inputs fades exponentially — fast and uniform.

The Fractional LIF (FLIF) Neuron

Motivation

The exponential decay of the classical LIF is at odds with biological observations. Cortical neurons exhibit power-law adaptation, ion channels display non-Markovian kinetics, and dendritic processing involves anomalous subdiffusion. All of these phenomena are naturally described by fractional-order dynamics.

The fractional LIF (FLIF) model replaces the integer-order time derivative with a Caputo fractional derivative of order $\alpha \in (0, 1]$.

Biophysical Form

Normalized Form

When $\alpha = 1$, the Caputo derivative reduces to the ordinary derivative and Equation (4) reduces to the classical LIF. When $\alpha \lt 1$, the neuron acquires a power-law memory kernel: the present voltage depends on the entire history of inputs and states, weighted by $(t - \tau)^{-\alpha}$.

Free Response and Mittag-Leffler Decay

For the unforced FLIF ($I(t) = 0$) with initial condition $V(0) = V_0$, the solution is:

where $E_\alpha(z) = \sum_{k=0}^\infty z^k / \Gamma(\alpha k + 1)$ is the Mittag-Leffler function. This function interpolates between two extremes:

• For $\alpha = 1$: $E_1(-t/\tau_m) = e^{-t/\tau_m}$ (exponential decay)
• For small $t$: $E_\alpha(-t^\alpha/\tau_m) \approx 1 - t^\alpha/(\tau_m \Gamma(\alpha + 1))$ (stretched exponential)
• For large $t$: $E_\alpha(-t^\alpha/\tau_m) \sim t^{-\alpha}/(\tau_m \Gamma(1 - \alpha))$ (power-law tail)

The power-law tail is the key feature. While the classical LIF forgets its initial condition exponentially fast, the FLIF retains a memory that decays only algebraically. This slow forgetting is precisely the behavior needed to capture long-range temporal dependencies.

Grunwald-Letnikov Discretization

Derivation

To simulate the FLIF numerically, we use the Grunwald-Letnikov (GL) discretization. Starting from the GL definition of the fractional derivative and setting $h = \Delta t$:

where $c_k(\alpha) = (-1)^k \binom{\alpha}{k}$ are the GL coefficients. Substituting into the FLIF equation (4) and solving for $V_n$:

Rearranging to isolate $V_n$:

This is the FLIF-GL update rule implemented in SPIRES. At each time step, the new voltage is computed from two contributions:

1. Instantaneous dynamics: The first term captures the leak and input at the current time step, scaled by $\Delta t^\alpha$.
2. History correction: The summation over past voltages, weighted by the GL coefficients, encodes the fractional memory.

GL Coefficient Computation

The GL coefficients are defined by the generalized binomial coefficients:

These can be computed efficiently via the recurrence:

The first several coefficients for representative values of $\alpha$:

Note that for $\alpha = 1$, only $c_0$ and $c_1$ are nonzero, and the update rule reduces to the classical Euler forward step. For $\alpha \lt 1$, the coefficients are nonzero for all $k$, encoding the infinite memory of the fractional operator.

History Length $L$

In principle, the GL sum extends over the entire history ($k = 0$ to $n$). In practice, the sum is truncated at a finite history length $L$. The truncation error decreases as:

The slower the power-law decay (smaller $\alpha$), the more history must be retained for a given accuracy. In SPIRES, $L$ is a configurable parameter. Typical values range from 50 to 500, depending on the task’s temporal scale and the chosen $\alpha$.

The computational cost of the history correction is $O(NL)$ per time step for a reservoir of $N$ neurons, making $L$ the primary knob for the memory-computation trade-off.

Effects of $\alpha$ on Neural Dynamics

The fractional order $\alpha$ profoundly shapes the behavior of the FLIF neuron across multiple dimensions.

Membrane Potential Decay

• $\alpha = 1.0$: Exponential decay with time constant $\tau_m$. The neuron “forgets” its state on a single timescale.
• $\alpha \lt 1.0$: Mittag-Leffler decay — initially stretched-exponential, asymptotically power-law. The neuron retains a fading trace of its entire history.

Effective Rheobase Shift

The rheobase is the minimum sustained current required to bring the neuron to threshold. For the classical LIF, the rheobase is:

For the FLIF, the fractional derivative introduces an effective increase in the rheobase. Intuitively, the power-law memory kernel causes the neuron to “remember” its subthreshold state more persistently, which opposes the charging process. Lower values of $\alpha$ produce a higher effective rheobase, meaning the neuron requires stronger input to fire.

This rheobase shift has an important consequence for reservoir computing: lower $\alpha$ makes the reservoir more selective, responding only to sufficiently strong or persistent input patterns.

Input Sensitivity vs. Memory Retention

The fractional order controls a fundamental trade-off between two desirable properties:

• Input sensitivity (high $\alpha$): The neuron responds rapidly to new inputs, making it an effective sensor of recent stimuli. However, it quickly forgets past events.
• Memory retention (low $\alpha$): The neuron maintains long traces of past inputs, enabling it to detect slow-varying patterns and long-range dependencies. However, it is less responsive to sudden changes.

This trade-off is not merely qualitative. It can be quantified precisely using information-theoretic measures, as discussed in Memory and Information Theory.

Frequency Response

The Laplace-domain transfer function of the FLIF neuron is:

This is a fractional-order low-pass filter. The roll-off rate is $-20\alpha$ dB/decade, compared to $-20$ dB/decade for the classical LIF. Lower $\alpha$ produces a shallower roll-off, meaning the FLIF neuron passes a broader range of frequencies — it is more sensitive to slow temporal components of the input.

Spike-and-Reset Mechanism

The spike-and-reset rule is the same for classical and fractional LIF neurons:

1. If $V_n \geq V_{\text{th}}$: record a spike at time $t_n$.
2. Set $V_n \leftarrow V_{\text{reset}}$.
3. Optionally, enforce a refractory period during which $V$ is held at $V_{\text{reset}}$.

An important subtlety arises with the GL discretization: when $V_n$ is reset, the history buffer retains the pre-reset voltage values. This means the fractional memory “remembers” the approach to threshold even after the reset, creating a form of spike aftereffect that influences subsequent dynamics. This is actually biologically realistic, as real neurons exhibit post-spike membrane potential trajectories that depend on the preceding interspike interval.

Summary

References

1. Teka, W. W., Marinov, T. M., & Bhatt, S. J. (2014). Fractional-order leaky integrate-and-fire model with long-term memory and power law dynamics. Computational and Mathematical Methods in Medicine, 2014.
2. Teka, W. W., Upadhyay, R. K., & Mondal, A. (2017). Fractional-order leaky integrate-and-fire model: frequency adaptation and coincidence detection. Biosystems, 155, 32—42.
3. Lundstrom, B. N., Higgs, M. H., Spain, W. J., & Fairhall, A. L. (2008). Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience, 11(11), 1335—1342.
4. Podlubny, I. (1999). Fractional Differential Equations. Academic Press.
5. Gorenflo, R., Kilbas, A. A., Mainardi, F., & Rogosin, S. V. (2014). Mittag-Leffler Functions, Related Topics and Applications. Springer.

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