Biophysical LIF Model

The Biophysical Leaky Integrate-and-Fire model provides a more physically grounded simulation of neuronal membrane dynamics. It uses an implicit Euler integration scheme that guarantees unconditional numerical stability, making it suitable for simulations where large time steps or stiff parameter regimes are required.

Continuous-Time Formulation

The biophysical LIF neuron is described by the membrane equation:

C \frac{dV}{dt} = -g_L\bigl(V(t) - E_L\bigr) + I(t)

where:

$C$ is the membrane capacitance (units: nF or pF)
$g_L$ is the leak conductance (units: nS or pS)
$E_L$ is the leak reversal potential (equivalent to $V_{\text{rest}}$ )
$I(t)$ is the total synaptic and external input current

The membrane time constant is $\tau_m = C / g_L$ . When no current is injected, the membrane potential decays exponentially toward $E_L$ with this time constant.

As with the discrete LIF, a spike is emitted when $V(t) \geq V_{\text{th}}$ , and the potential is immediately reset to $V_{\text{reset}}$ .

Implicit Euler Discretization

The biophysical LIF model in SPIRES uses the implicit (backward) Euler method. Instead of evaluating the right-hand side at the current time step $n-1$ (as in forward Euler), implicit Euler evaluates it at the next time step $n$ :

C \frac{V_n - V_{n-1}}{\Delta t} = -g_L\bigl(V_n - E_L\bigr) + I_n

Solving for $V_n$ :

V_n = \frac{C\,V_{n-1} + \Delta t\bigl(g_L\,E_L + I_n\bigr)}{C + \Delta t\,g_L}

Or equivalently, in terms of the membrane time constant $\tau_m = C / g_L$ :

V_n = \frac{\tau_m\,V_{n-1} + \Delta t\bigl(E_L + I_n / g_L\bigr)}{\tau_m + \Delta t}

This formula requires no iterative solve — it is an explicit expression for $V_n$ despite being derived from the implicit method. The implicit character manifests in the denominator $(\tau_m + \Delta t)$ , which acts as a stabilizing divisor that prevents runaway growth regardless of the time step size.

Unconditional Stability

The critical advantage of implicit Euler is unconditional stability. The amplification factor for the homogeneous equation ( $I = 0$ ) is:

\left|\frac{V_n}{V_{n-1}}\right| = \frac{\tau_m}{\tau_m + \Delta t} < 1 \quad \text{for all } \Delta t > 0

This means the membrane potential always decays toward the resting potential, no matter how large $\Delta t$ is. Compare this with forward Euler, which requires $\Delta t \lt 2\tau_m$ for stability.

In practice, unconditional stability is valuable when:

You want to use large time steps for computational efficiency
You are sweeping parameters across wide ranges during optimization
You need reliable behavior in automated hyperparameter searches where parameter combinations may be extreme

Parameters

Parameter	Symbol	Description	Typical Range
Membrane capacitance	$C$	Total membrane capacitance	0.1—1.0 nF
Leak conductance	$g_L$	Passive leak conductance	10—50 nS
Leak reversal potential	$E_L$	Resting equilibrium potential	-70 to -60 mV
Threshold	$V_{\text{th}}$	Spike initiation threshold	-55 to -45 mV
Reset potential	$V_{\text{reset}}$	Post-spike reset value	-70 to -60 mV

The effective membrane time constant is derived as $\tau_m = C / g_L$ . For $C = 0.5$ nF and $g_L = 25$ nS, $\tau_m = 20$ ms.

SPIRES API

The biophysical LIF model is selected with:


cfg.neuron_type = SPIRES_NEURON_LIF_BIO;

Example Configuration


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                = 0.5,
    .connectivity_type = SPIRES_CONN_RANDOM,
    .neuron_type       = SPIRES_NEURON_LIF_BIO,
    .neuron_params     = NULL,  /* use defaults */
};

Custom Parameters


spires_lif_bio_params params = {
    .C       = 0.5,     /* nF */
    .g_L     = 25.0,    /* nS */
    .E_L     = -65.0,   /* mV */
    .v_th    = -50.0,   /* mV */
    .v_reset = -65.0,   /* mV */
};
cfg.neuron_params = &params;

As with all neuron parameter structs, the caller owns the memory. SPIRES copies the values during spires_reservoir_create().

Biological Motivation

The conductance-based formulation used by the biophysical LIF is closer to how real neurons operate. In biological neurons, the membrane is a capacitor (the lipid bilayer) with parallel conductance channels (ion channels). The leak conductance $g_L$ represents the sum of passive ion channels that are always open.

While the biophysical LIF is still a significant simplification of real neuronal dynamics (it omits voltage-dependent ion channels, synaptic dynamics, and dendritic processing), it captures the essential RC-circuit behavior of the neuronal membrane more faithfully than the simplified discrete LIF formulation.

When to Use Biophysical LIF

Choose the biophysical LIF when:

Stability is paramount: Automated sweeps, optimization runs, or production pipelines where numerical blowup would be costly.
Large time steps are needed: When you need to simulate long time spans efficiently.
Biophysical parameters are meaningful: When you want to specify capacitance and conductance directly, rather than abstract time constants.
Comparing with neuroscience models: When your reservoir computing results will be interpreted in a neuroscience context.

For the fastest possible execution with small time steps, the Discrete LIF remains slightly cheaper per step. For problems requiring long memory, consider the fractional LIF models.

← Discrete LIF | Caputo Fractional LIF →