glif_cond

glif_cond#

class brainpy.state.glif_cond(in_size, g=Quantity(9.43, 'nS'), E_L=Quantity(-78.85, 'mV'), V_th=Quantity(-51.68, 'mV'), C_m=Quantity(58.72, 'pF'), t_ref=Quantity(3.75, 'ms'), V_reset=Quantity(-78.85, 'mV'), th_spike_add=0.37, th_spike_decay=0.009, voltage_reset_fraction=0.2, voltage_reset_add=18.51, th_voltage_index=0.005, th_voltage_decay=0.09, asc_init=(0.0, 0.0), asc_decay=(0.003, 0.1), asc_amps=(-9.18, -198.94), asc_r=(1.0, 1.0), tau_syn=(0.2, 2.0), E_rev=(0.0, -85.0), spike_dependent_threshold=False, after_spike_currents=False, adapting_threshold=False, I_e=Quantity(0., 'pA'), V_initializer=None, spk_fun=ReluGrad(alpha=0.3, width=1.0), spk_reset='hard', name=None)#

Conductance-based generalized leaky integrate-and-fire (GLIF) neuron model.

Implements the five-level GLIF model hierarchy from Teeter et al. (2018) [1], with conductance-based alpha-function synapses and adaptive RKF45 integration. Designed for fitting to Allen Institute single-neuron electrophysiology data. Supports multiple receptor ports with distinct reversal potentials and synaptic time constants.

Model Selection

The five GLIF variants are:

GLIF1 (LIF) — Traditional leaky integrate-and-fire
GLIF2 (LIF_R) — LIF with biologically defined voltage reset rules
GLIF3 (LIF_ASC) — LIF with after-spike currents (adaptation)
GLIF4 (LIF_R_ASC) — LIF with reset rules and after-spike currents
GLIF5 (LIF_R_ASC_A) — LIF with reset rules, after-spike currents, and voltage-dependent threshold

Model mechanism selection is controlled by three boolean parameters:

Model	spike_dependent_threshold	after_spike_currents	adapting_threshold
GLIF1	False	False	False
GLIF2	True	False	False
GLIF3	False	True	False
GLIF4	True	True	False
GLIF5	True	True	True

Mathematical Formulation

1. Membrane Dynamics

The membrane potential \(V\) (tracked relative to \(E_L\) internally) evolves according to:

\[C_\mathrm{m} \frac{dV}{dt} = -g \cdot V - \sum_k g_k(t) \left( V + E_L - E_{\mathrm{rev},k} \right) + I_\mathrm{e} + I_\mathrm{ASC,sum}\]

where:

\(g\) — membrane (leak) conductance
\(g_k(t)\) — synaptic conductance for receptor port \(k\)
\(E_{\mathrm{rev},k}\) — reversal potential for port \(k\)
\(I_\mathrm{e}\) — constant external current
\(I_\mathrm{ASC,sum}\) — sum of after-spike currents (GLIF3/4/5 only)

2. Synaptic Conductances (Alpha Function)

Each receptor port \(k\) has a conductance modeled by an alpha function with two state variables \(dg_k\) and \(g_k\):

\[\frac{d(dg_k)}{dt} = -\frac{dg_k}{\tau_{\mathrm{syn},k}}\]

\[\frac{dg_k}{dt} = dg_k - \frac{g_k}{\tau_{\mathrm{syn},k}}\]

On a presynaptic spike with weight \(w\), the derivative is incremented:

\[dg_k \leftarrow dg_k + w \cdot \frac{e}{\tau_{\mathrm{syn},k}}\]

This normalization ensures that a spike of weight 1.0 produces a peak conductance of 1 nS at time \(t = \tau_{\mathrm{syn},k}\).

3. After-Spike Currents (GLIF3/4/5)

After-spike currents (ASC) model spike-triggered adaptation as exponentially decaying currents. Each ASC component \(I_j\) decays with rate \(k_j\):

\[I_j(t+dt) = I_j(t) \cdot \exp(-k_j \cdot dt)\]

The time-averaged ASC over a simulation step uses the exact integral (stable coefficient method):

\[\bar{I}_j = \frac{1 - \exp(-k_j \cdot dt)}{k_j \cdot dt} \cdot I_j(t)\]

On spike, ASC values are updated with amplitude and refractory decay:

\[I_j \leftarrow \Delta I_j + I_j \cdot r_j \cdot \exp(-k_j \cdot t_\mathrm{ref})\]

where \(\Delta I_j\) is the amplitude jump and \(r_j \in [0, 1]\) is the retention fraction.

4. Spike-Dependent Threshold (GLIF2/4/5)

The spike component of the threshold \(\theta_s\) decays exponentially:

\[\theta_s(t+dt) = \theta_s(t) \cdot \exp(-b_s \cdot dt)\]

On spike, after accounting for refractory decay, it is incremented:

\[\theta_s \leftarrow \theta_s \cdot \exp(-b_s \cdot t_\mathrm{ref}) + \Delta\theta_s\]

Voltage reset with spike-dependent threshold uses:

\[V \leftarrow f_v \cdot V_\mathrm{old} + V_\mathrm{add}\]

where \(f_v \in [0, 1]\) is the fraction coefficient and \(V_\mathrm{add}\) is the additive term (both in mV, dimensionless in NEST convention).

5. Voltage-Dependent Threshold (GLIF5)

The voltage component \(\theta_v\) evolves according to:

\[\theta_v(t+dt) = \phi \cdot (V_\mathrm{old} - \beta) \cdot P_\mathrm{decay} + \frac{1}{P_{\theta,v}} \cdot \left(\theta_v(t) - \phi \cdot (V_\mathrm{old} - \beta) - \frac{a_v}{b_v} \cdot \beta \right) + \frac{a_v}{b_v} \cdot \beta\]

where:

\(\phi = a_v / (b_v - g/C_m)\)
\(P_\mathrm{decay} = \exp(-g \cdot dt / C_m)\)
\(P_{\theta,v} = \exp(b_v \cdot dt)\)
\(\beta = (I_e + I_\mathrm{ASC,sum}) / g\)

The total threshold is the sum of all components:

\[\theta = \theta_\infty + \theta_s + \theta_v\]

Spike condition (checked after ODE integration):

\[V > \theta\]

Numerical Integration

The ODE system \([V, dg_0, g_0, dg_1, g_1, \ldots]\) is integrated using an adaptive RKF45(4,5) Runge-Kutta-Fehlberg method with error tolerance ATOL = 1e-3 and minimum step size MIN_H = 1e-8 ms, matching NEST’s GSL integrator behavior.

Update Order (Per Simulation Step)

Record \(V_\mathrm{old}\) (relative to \(E_L\))
Integrate ODE system over \((t, t+dt]\) using RKF45
If not refractory:
1. Decay spike threshold component \(\theta_s\)
2. Compute time-averaged ASC \(\bar{I}_\mathrm{ASC,sum}\) and decay ASC values
3. Compute voltage-dependent threshold \(\theta_v\) (using \(V_\mathrm{old}\))
4. Update total threshold \(\theta = \theta_\infty + \theta_s + \theta_v\)
5. If \(V > \theta\): emit spike, apply reset rules
If refractory: decrement counter, clamp \(V\) to \(V_\mathrm{old}\)
Add incoming spike conductance jumps (scaled by \(e/\tau_\mathrm{syn}\))
Update external current buffer \(I_\mathrm{stim}\)
Save \(V_\mathrm{old}\) for next step

Parameters:

in_size (Size) – Shape of the neuron population. Can be an int for 1D or tuple for multi-D.
g (ArrayLike, optional) – Membrane (leak) conductance in nS. Broadcast to population shape. Default: 9.43 nS (from Allen Cell 490626718 GLIF5).
E_L (ArrayLike, optional) – Resting membrane potential (leak reversal) in mV. Default: -78.85 mV.
V_th (ArrayLike, optional) – Instantaneous spike threshold (absolute) in mV. Default: -51.68 mV. Internally, threshold is tracked relative to E_L.
C_m (ArrayLike, optional) – Membrane capacitance in pF. Must be strictly positive. Default: 58.72 pF.
t_ref (ArrayLike, optional) – Absolute refractory period in ms. During this period, voltage is clamped and spike detection is disabled. Must be > 0. Default: 3.75 ms.
V_reset (ArrayLike, optional) – Reset potential (absolute) in mV for GLIF1/3 models. Ignored if spike_dependent_threshold=True. Default: -78.85 mV (same as E_L).
th_spike_add (float, optional) – Threshold additive constant \(\Delta\theta_s\) after spike (mV, dimensionless in NEST units). Only used if spike_dependent_threshold=True. Default: 0.37 mV.
th_spike_decay (float, optional) – Spike threshold decay rate \(b_s\) in 1/ms. Must be > 0 if spike_dependent_threshold=True. Default: 0.009 /ms.
voltage_reset_fraction (float, optional) – Voltage fraction coefficient \(f_v \in [0, 1]\) after spike. Only used if spike_dependent_threshold=True. Default: 0.20.
voltage_reset_add (float, optional) – Voltage additive term \(V_\mathrm{add}\) after spike (mV, dimensionless). Only used if spike_dependent_threshold=True. Default: 18.51 mV.
th_voltage_index (float, optional) – Voltage-dependent threshold leak \(a_v\) in 1/ms. Only used if adapting_threshold=True. Default: 0.005 /ms.
th_voltage_decay (float, optional) – Voltage-dependent threshold decay rate \(b_v\) in 1/ms. Must be > 0 if adapting_threshold=True. Default: 0.09 /ms.
asc_init (Sequence[float], optional) – Initial values of after-spike currents in pA. Tuple/list of length n_asc. Default: (0.0, 0.0) pA.
asc_decay (Sequence[float], optional) – ASC decay rates \(k_j\) in 1/ms. All values must be > 0. Length must match asc_init. Default: (0.003, 0.1) /ms.
asc_amps (Sequence[float], optional) – ASC amplitude jumps \(\Delta I_j\) on spike, in pA. Length must match asc_init. Negative values cause hyperpolarizing adaptation. Default: (-9.18, -198.94) pA.
asc_r (Sequence[float], optional) – ASC retention fraction coefficients \(r_j \in [0, 1]\). Length must match asc_init. Default: (1.0, 1.0).
tau_syn (Sequence[float], optional) – Synaptic alpha-function time constants \(\tau_{\mathrm{syn},k}\) in ms, one per receptor port. All values must be > 0. Default: (0.2, 2.0) ms (fast excitatory, slow inhibitory).
E_rev (Sequence[float], optional) – Synaptic reversal potentials \(E_{\mathrm{rev},k}\) in mV, one per receptor port. Must have same length as tau_syn. Default: (0.0, -85.0) mV (excitatory, inhibitory).
spike_dependent_threshold (bool, optional) – Enable biologically defined voltage reset rules (GLIF2/4/5). Default: False.
after_spike_currents (bool, optional) – Enable after-spike currents (adaptation) (GLIF3/4/5). Default: False.
adapting_threshold (bool, optional) – Enable voltage-dependent threshold component (GLIF5 only). Requires spike_dependent_threshold=True and after_spike_currents=True. Default: False.
I_e (ArrayLike, optional) – Constant external current in pA. Broadcast to population shape. Default: 0.0 pA.
V_initializer (Callable, optional) – Initializer for membrane potential. If None, defaults to Constant(E_L).
spk_fun (Callable, optional) – Surrogate gradient function for spike generation. Default: ReluGrad().
spk_reset (str, optional) – Spike reset mode: 'hard' (stop gradient) or 'soft' (subtract threshold). Default: 'hard'.
name (str, optional) – Name of the neuron population.

Parameter Mapping

Parameter	Default	Math equivalent	Description
`in_size`	(required)		Population shape
`g`	9.43 nS	\(g\)	Membrane (leak) conductance
`E_L`	-78.85 mV	\(E_L\)	Resting membrane potential
`V_th`	-51.68 mV	\(V_\mathrm{th}\)	Instantaneous threshold (absolute)
`C_m`	58.72 pF	\(C_\mathrm{m}\)	Membrane capacitance
`t_ref`	3.75 ms	\(t_\mathrm{ref}\)	Absolute refractory period
`V_reset`	-78.85 mV	\(V_\mathrm{reset}\)	Reset potential (absolute; GLIF1/3)
`th_spike_add`	0.37 mV	\(\Delta\theta_s\)	Threshold additive constant after spike
`th_spike_decay`	0.009 /ms	\(b_s\)	Spike threshold decay rate
`voltage_reset_fraction`	0.20	\(f_v\)	Voltage fraction after spike
`voltage_reset_add`	18.51 mV	\(V_\mathrm{add}\)	Voltage additive after spike
`th_voltage_index`	0.005 /ms	\(a_v\)	Voltage-dependent threshold leak
`th_voltage_decay`	0.09 /ms	\(b_v\)	Voltage-dependent threshold decay rate
`asc_init`	(0.0, 0.0) pA		Initial values of ASC
`asc_decay`	(0.003, 0.1) /ms	\(k_j\)	ASC time constants (decay rates)
`asc_amps`	(-9.18, -198.94) pA	\(\Delta I_j\)	ASC amplitudes on spike
`asc_r`	(1.0, 1.0)	\(r_j\)	ASC fraction coefficient
`tau_syn`	(0.2, 2.0) ms	\(\tau_{\mathrm{syn},k}\)	Synaptic alpha-function time constants
`E_rev`	(0.0, -85.0) mV	\(E_{\mathrm{rev},k}\)	Synaptic reversal potentials
`spike_dependent_threshold`	False		Enable biologically defined reset (GLIF2/4/5)
`after_spike_currents`	False		Enable after-spike currents (GLIF3/4/5)
`adapting_threshold`	False		Enable voltage-dependent threshold (GLIF5)
`I_e`	0.0 pA	\(I_e\)	Constant external current
`V_initializer`	Constant(E_L)		Membrane potential initializer
`spk_fun`	ReluGrad()		Surrogate spike function
`spk_reset`	`'hard'`		Reset mode

V#

Membrane potential in mV (absolute, broadcast to population shape).

Type:: HiddenState

g_syn#

Synaptic conductances \(g_k\) in nS, one per receptor port.

Type:: list[HiddenState]

dg_syn#

Synaptic conductance derivatives \(dg_k\) in nS, one per receptor port.

Type:: list[HiddenState]

last_spike_time#

Time of last spike in ms.

Type:: ShortTermState

refractory_step_count#

Remaining refractory steps (int32), decremented each step.

Type:: ShortTermState

integration_step#

Internal RKF45 adaptive step size in ms (updated per neuron).

Type:: ShortTermState

I_stim#

Buffered external current in pA (applied with one-step delay).

Type:: ShortTermState

Notes

Implementation Details

Internal state convention: Membrane potential is tracked relative to E_L internally (matching NEST), but exposed as absolute values in mV.
Threshold components: _threshold_spike, _threshold_voltage, and _th_inf are stored as numpy arrays (not JAX) for exact NEST replication.
After-spike currents: _ASCurrents is a numpy array of shape (n_asc, *in_size).
Receptor port routing: Delta inputs (from projections) with keys containing 'receptor_<k>' (0-based) are routed to receptor port k. Inputs without a receptor tag default to receptor 0.
Stability constraint: For GLIF2/4/5, the reset condition must satisfy:

\[E_L + f_v \cdot (V_\mathrm{th} - E_L) + V_\mathrm{add} < V_\mathrm{th} + \Delta\theta_s\]

Otherwise the neuron may spike continuously.
Valid mechanism combinations: Only the five combinations listed in the parameter table are valid. Other combinations will raise ValueError.
Adaptive integration: RKF45 step size adapts per-neuron and is preserved across simulation steps.

Failure Modes

Raises ValueError if parameter validation fails (invalid model combination, non-positive capacitance/conductance/time constants, mismatched sequence lengths).
Raises ValueError if V_reset >= V_th (relative to E_L).
Integration may fail to converge if dt is too large relative to tau_syn or if threshold parameters cause continuous spiking.

Default Parameters

Default parameter values are from GLIF Model 5 of Cell 490626718 from the Allen Cell Type Database, fitted to mouse visual cortex layer 5 pyramidal neuron electrophysiology.

References

Examples

Example 1: GLIF1 (simple LIF) with dual-receptor synapses

>>> import brainpy.state as bst
>>> import saiunit as u
>>> import brainstate as bts
>>> # Create GLIF1 neuron (all mechanisms disabled)
>>> neuron = bst.glif_cond(
...     100,
...     spike_dependent_threshold=False,
...     after_spike_currents=False,
...     adapting_threshold=False,
...     tau_syn=(0.2, 2.0),  # fast excitatory, slow inhibitory
...     E_rev=(0.0, -85.0)   # mV
... )
>>> neuron.init_all_states()
>>> # Stimulate with constant current
>>> with bts.environ.context(dt=0.1 * u.ms):
...     for _ in range(100):
...         spike = neuron(200.0 * u.pA)

Example 2: GLIF5 (full model) with custom parameters

>>> # Create GLIF5 with all mechanisms enabled
>>> neuron = bst.glif_cond(
...     (10, 10),  # 10x10 population
...     spike_dependent_threshold=True,
...     after_spike_currents=True,
...     adapting_threshold=True,
...     g=10.0 * u.nS,
...     C_m=100.0 * u.pF,
...     tau_syn=(0.5, 1.5, 5.0),  # three receptor ports
...     E_rev=(0.0, 0.0, -80.0)   # two excitatory, one inhibitory
... )
>>> neuron.init_all_states()
>>> print(neuron.n_receptors)  # 3

Example 3: Multi-receptor input routing

>>> from brainevent.nn import FixedProb
>>> # Create projection targeting receptor 1
>>> proj = bst.align_post_projection(
...     pre=pre_neurons,
...     post=glif_neurons,
...     comm=FixedProb(0.1, weight=0.5 * u.nS),
...     label='receptor_1'  # route to receptor port 1
... )

glif_cond

Contents

glif_cond#