iaf_cond_alpha

class brainpy.state.iaf_cond_alpha(in_size, E_L=Quantity(-70., "mV"), C_m=Quantity(250., "pF"), t_ref=Quantity(2., "ms"), V_th=Quantity(-55., "mV"), V_reset=Quantity(-60., "mV"), E_ex=Quantity(0., "mV"), E_in=Quantity(-85., "mV"), g_L=Quantity(16.6667, "nS"), tau_syn_ex=Quantity(0.2, "ms"), tau_syn_in=Quantity(2., "ms"), I_e=Quantity(0., "pA"), gsl_error_tol=0.001, V_initializer=Constant(value=-70. mV), g_ex_initializer=Constant(value=0. nS), g_in_initializer=Constant(value=0. nS), spk_fun=ReluGrad(alpha=0.3, width=1.0), spk_reset='hard', ref_var=False, name=None)

Leaky integrate-and-fire model with alpha-shaped conductance synapses.

Description

iaf_cond_alpha is a conductance-based leaky integrate-and-fire neuron with

• hard threshold,

• fixed absolute refractory period,

• alpha-shaped excitatory and inhibitory synaptic conductances (second-order kinetics),

• no adaptation variables.

This implementation follows NEST iaf_cond_alpha dynamics and update order, using NEST C++ model behavior as the source of truth.

1. Membrane Potential and Synaptic Currents

The membrane potential evolves according to

\[\frac{dV_\mathrm{m}}{dt} = \frac{-g_\mathrm{L}(V_\mathrm{m}-E_\mathrm{L}) - I_\mathrm{syn} + I_\mathrm{e} + I_\mathrm{stim}} {C_\mathrm{m}}\]

with

\[I_\mathrm{syn} = I_{\mathrm{syn,ex}} + I_{\mathrm{syn,in}} = g_\mathrm{ex}(V_\mathrm{m}-E_\mathrm{ex}) + g_\mathrm{in}(V_\mathrm{m}-E_\mathrm{in}) .\]

2. Alpha-Shaped Conductance Kinetics

Alpha conductances use two coupled state variables per channel:

\[\frac{d\,dg_\mathrm{ex}}{dt} = -\frac{dg_\mathrm{ex}}{\tau_{\mathrm{syn,ex}}}, \qquad \frac{d g_\mathrm{ex}}{dt} = dg_\mathrm{ex} - \frac{g_\mathrm{ex}}{\tau_{\mathrm{syn,ex}}},\]

\[\frac{d\,dg_\mathrm{in}}{dt} = -\frac{dg_\mathrm{in}}{\tau_{\mathrm{syn,in}}}, \qquad \frac{d g_\mathrm{in}}{dt} = dg_\mathrm{in} - \frac{g_\mathrm{in}}{\tau_{\mathrm{syn,in}}}.\]

A presynaptic spike with weight \(w\) causes an instantaneous jump at the end of the simulation step. Positive/negative weights map to excitatory/inhibitory channels:

\[w > 0 \Rightarrow dg_\mathrm{ex} \leftarrow dg_\mathrm{ex} + \frac{e}{\tau_{\mathrm{syn,ex}}} w,\]

\[w < 0 \Rightarrow dg_\mathrm{in} \leftarrow dg_\mathrm{in} + \frac{e}{\tau_{\mathrm{syn,in}}} |w|.\]

The normalization factor \(e/\tau\) ensures the conductance peak matches the weight magnitude (in nS).

3. Spike Emission and Refractory Mechanism

A spike is emitted when \(V_\mathrm{m} \ge V_\mathrm{th}\) at the end of a simulation step. On spike:

• \(V_\mathrm{m}\) is reset to \(V_\mathrm{reset}\),

• refractory counter is set to \(\lceil t_\mathrm{ref}/dt \rceil\),

• spike time is recorded as \(t + dt\).

During absolute refractory period:

• effective membrane potential in current computation is clamped to \(V_\mathrm{reset}\),

• \(dV_\mathrm{m}/dt = 0\),

• conductances continue to decay.

4. Numerical Integration and Update Order

NEST integrates this model with adaptive RKF45. This implementation mirrors that behavior with an RKF45(4,5) integrator and persistent internal step size. The discrete-time update order is:

1. Integrate continuous dynamics on \((t, t+dt]\) using RKF45 with adaptive substeps.

2. Apply refractory countdown / threshold test / reset and spike emission.

3. Add synaptic conductance jumps from spike inputs arriving this step.

4. Store external current input as \(I_\mathrm{stim}\) for the next step.

The one-step delayed application of current input (I_stim buffer) is intentional and matches NEST’s ring-buffer update semantics.

Parameters:

• in_size (tuple of int or int) – Shape of the neuron population. Can be an integer for 1D populations or a tuple for multi-dimensional populations.

• E_L (ArrayLike, optional) – Leak reversal potential \(E_\mathrm{L}\). Default: -70 mV.

• C_m (ArrayLike, optional) – Membrane capacitance \(C_\mathrm{m}\). Must be strictly positive. Default: 250 pF.

• t_ref (ArrayLike, optional) – Absolute refractory period \(t_\mathrm{ref}\). Must be non-negative. Default: 2 ms.

• V_th (ArrayLike, optional) – Spike threshold \(V_\mathrm{th}\). Must be larger than V_reset. Default: -55 mV.

• V_reset (ArrayLike, optional) – Reset potential \(V_\mathrm{reset}\). Must be smaller than V_th. Default: -60 mV.

• E_ex (ArrayLike, optional) – Excitatory reversal potential \(E_\mathrm{ex}\). Default: 0 mV.

• E_in (ArrayLike, optional) – Inhibitory reversal potential \(E_\mathrm{in}\). Default: -85 mV.

• g_L (ArrayLike, optional) – Leak conductance \(g_\mathrm{L}\). Must be strictly positive. Default: 16.6667 nS (yields \(\tau_\mathrm{m} = 15\) ms with default C_m).

• tau_syn_ex (ArrayLike, optional) – Excitatory alpha time constant \(\tau_{\mathrm{syn,ex}}\). Must be strictly positive. Default: 0.2 ms.

• tau_syn_in (ArrayLike, optional) – Inhibitory alpha time constant \(\tau_{\mathrm{syn,in}}\). Must be strictly positive. Default: 2.0 ms.

• I_e (ArrayLike, optional) – Constant external current \(I_\mathrm{e}\). Default: 0 pA.

• gsl_error_tol (ArrayLike, optional) – Unitless local RKF45 error tolerance, broadcastable and strictly positive. Default: 1e-3.

• V_initializer (Callable, optional) – Initializer for membrane potential. Default: Constant(-70 mV).

• g_ex_initializer (Callable, optional) – Initializer for excitatory conductance. Default: Constant(0 nS).

• g_in_initializer (Callable, optional) – Initializer for inhibitory conductance. Default: Constant(0 nS).

• spk_fun (Callable, optional) – Surrogate gradient function for spike generation (differentiable approximation). Default: ReluGrad().

• spk_reset (str, optional) – Spike reset mode. 'hard' uses stop_gradient (matches NEST behavior), 'soft' allows gradients through reset. Default: 'hard'.

• ref_var (bool, optional) – If True, expose refractory state variable as boolean indicator. Default: False.

• name (str, optional) – Name of the neuron group.

Parameter Mapping

Parameter	Default	Math equivalent
in_size	(required)	–
E_L	-70 mV	\(E_\mathrm{L}\)
C_m	250 pF	\(C_\mathrm{m}\)
t_ref	2 ms	\(t_\mathrm{ref}\)
V_th	-55 mV	\(V_\mathrm{th}\)
V_reset	-60 mV	\(V_\mathrm{reset}\)
E_ex	0 mV	\(E_\mathrm{ex}\)
E_in	-85 mV	\(E_\mathrm{in}\)
g_L	16.6667 nS	\(g_\mathrm{L}\)
tau_syn_ex	0.2 ms	\(\tau_{\mathrm{syn,ex}}\)
tau_syn_in	2.0 ms	\(\tau_{\mathrm{syn,in}}\)
I_e	0 pA	\(I_\mathrm{e}\)
gsl_error_tol	1e-3	–
V_initializer	Constant(-70 mV)	–
g_ex_initializer	Constant(0 nS)	–
g_in_initializer	Constant(0 nS)	–
spk_fun	ReluGrad()	–
spk_reset	'hard'	–
ref_var	False	–

State Variables

State variable	Description
V	Membrane potential \(V_\mathrm{m}\)
dg_ex	Excitatory alpha auxiliary state
g_ex	Excitatory conductance \(g_\mathrm{ex}\)
dg_in	Inhibitory alpha auxiliary state
g_in	Inhibitory conductance \(g_\mathrm{in}\)
last_spike_time	Last spike time (recorded at \(t+dt\))
refractory_step_count	Remaining refractory grid steps
integration_step	Internal RKF45 step-size state (persistent)
I_stim	Buffered current applied in next step
refractory	Optional boolean refractory indicator (if ref_var=True)

Sends: SpikeEvent (conceptually; represented as returned spike tensor from update).

Receives: Signed spike-weight conductance increments through add_delta_input. - External current input through x in update() (one-step delayed).

Raises:

ValueError – If V_reset >= V_th, C_m <= 0, t_ref < 0, or any time constants are non-positive.

Notes

• Defaults follow NEST C++ source for iaf_cond_alpha (models/iaf_cond_alpha.h/.cpp).

• Synaptic spike weights are interpreted in conductance units (nS), with positive/negative sign selecting excitatory/inhibitory channel.

• The alpha shape produces a smoother conductance transient than single exponentials, peaking at \(t = \tau\) after a spike.

• During refractory period, the effective voltage used for current computation is clamped, but the actual V state continues to be updated (remains at reset value).

References

[1]

Meffin H, Burkitt AN, Grayden DB (2004). An analytical model for the large, fluctuating synaptic conductance state typical of neocortical neurons in vivo. Journal of Computational Neuroscience, 16:159-175. DOI: https://doi.org/10.1023/B:JCNS.0000014108.03012.81

[2]

Bernander O, Douglas RJ, Martin KAC, Koch C (1991). Synaptic background activity influences spatiotemporal integration in single pyramidal cells. PNAS, 88(24):11569-11573. DOI: https://doi.org/10.1073/pnas.88.24.11569

[3]

Kuhn A, Rotter S (2004). Neuronal integration of synaptic input in the fluctuation-driven regime. Journal of Neuroscience, 24(10):2345-2356. DOI: https://doi.org/10.1523/JNEUROSCI.3349-03.2004

[4]

NEST Simulator iaf_cond_alpha model documentation and C++ source: models/iaf_cond_alpha.h and models/iaf_cond_alpha.cpp.

See also

iaf_cond_exp

Conductance-based LIF with exponential synapses

iaf_psc_alpha

Current-based LIF with alpha synapses

iaf_psc_delta

Current-based LIF with delta synapses

Examples

Create a population of 100 conductance-based neurons:

>>> import brainpy.state as bst
>>> import saiunit as u
>>> neurons = bst.iaf_cond_alpha(
...     in_size=100,
...     V_th=-50. * u.mV,
...     tau_syn_ex=0.5 * u.ms,
...     tau_syn_in=2.0 * u.ms
... )

get_spike(V=None)

Compute differentiable spike output using surrogate gradient.

Applies the surrogate spike function to a normalized voltage to produce a continuous approximation of spike events suitable for gradient-based learning.

Parameters:

V (ArrayLike, optional) – Membrane potential to evaluate. If None, uses current self.V.value. Shape must match neuron population shape.

Returns:

Spike output in [0, 1], where values close to 1 indicate spike events. Shape matches input voltage shape.

Return type:

ArrayLike

Notes

The voltage is normalized to \((V - V_\mathrm{th}) / (V_\mathrm{th} - V_\mathrm{reset})\) before applying the surrogate function. This makes the surrogate function operate in a standardized range regardless of absolute voltage values.

init_state(**kwargs)

Initialize all state variables.

Creates and initializes membrane potential, conductance states, refractory counters, integration step size, and optional refractory indicator.

Parameters:

**kwargs (dict) – Additional keyword arguments (unused, for API compatibility).

Notes

• V, g_ex, g_in are initialized using their respective initializers.

• dg_ex, dg_in (alpha auxiliary states) are initialized to zero.

• last_spike_time is set to large negative value (-1e7 ms).

• refractory_step_count starts at 0 (not in refractory period).

• integration_step is initialized to the global timestep dt.

• I_stim buffer starts at 0 pA.

update(x=Quantity(0., 'pA'))

Advance neuron state by one simulation timestep.

Integrates ODEs, handles refractory period and spike emission, applies synaptic conductance jumps, and buffers external current for next step. This method implements the full NEST update semantics.

Parameters:

x (ArrayLike, optional) – External current input for this timestep (pA). Broadcasted to population shape. This input is buffered and applied in the next timestep (one-step delay) to match NEST ring-buffer semantics. Default: 0 pA.

Returns:

Differentiable spike output (values in [0, 1], shape matching population). Computed using surrogate gradient on pre-reset membrane potential.

Return type:

ArrayLike

Notes

Update order (matching NEST):

1. Integrate ODEs: Use RKF45 to advance V, dg_ex, g_ex, dg_in, g_in over (t, t+dt] with I_stim from previous step.

2. Refractory/spike handling:

   • If in refractory period: clamp V to V_reset, decrement counter.

   • Else if V >= V_th: emit spike, reset V to V_reset, set refractory counter.

3. Apply synaptic inputs: Add conductance jumps from incoming spikes (via add_delta_input) to dg_ex / dg_in with alpha normalization.

4. Buffer current input: Store x into I_stim for next timestep.

The surrogate spike is computed from the pre-reset voltage to allow gradient flow through spike events during training.

Failure modes: If integration does not converge within _MAX_ITERS iterations, the final state may be inaccurate. Reduce global dt or check for extreme parameter values if this occurs.