iaf_psc_delta

class brainpy.state.iaf_psc_delta(in_size, E_L=Quantity(-70., "mV"), C_m=Quantity(250., "pF"), tau_m=Quantity(10., "ms"), t_ref=Quantity(2., "ms"), V_th=Quantity(-55., "mV"), V_reset=Quantity(-70., "mV"), I_e=Quantity(0., "pA"), V_min=None, V_initializer=Constant(value=-70. mV), spk_fun=ReluGrad(alpha=0.3, width=1.0), spk_reset='hard', refractory_input=False, ref_var=False, name=None)

NEST-compatible iaf_psc_delta neuron model.

Description

iaf_psc_delta is a current-based leaky integrate-and-fire neuron with hard threshold/reset, absolute refractory period, and delta-shaped synaptic events represented as instantaneous membrane-voltage jumps (weights in mV). The implementation follows the NEST model iaf_psc_delta update semantics, including refractory handling and step-wise exact subthreshold propagation.

1. Continuous-Time Dynamics and Exact Per-Step Propagator

The membrane dynamics are

\[\frac{dV_\text{m}}{dt} = -\frac{V_{\text{m}} - E_\text{L}}{\tau_{\text{m}}} + \dot{\Delta}_{\text{syn}} + \frac{I_{\text{syn}} + I_\text{e}}{C_{\text{m}}}\]

where \(I_\text{syn}\) is the sum of continuous current inputs and \(\dot{\Delta}_{\text{syn}}\) captures impulse-like jumps from delta synapses.

For fixed simulation step \(h=dt\) and piecewise-constant current \(I_k\), exact integration of the linear subthreshold ODE yields

\[V_{k+1}^{\mathrm{cand}} = E_L + (V_k - E_L)e^{-h/\tau_m} + \frac{\tau_m}{C_m}\left(I_k + I_e\right)\left(1 - e^{-h/\tau_m}\right),\]

which is implemented directly in update(). This is equivalent to the propagator formulation used in NEST for this linear system.

2. Spike Condition, Reset, and Refractory Countdown

After adding delta-input jump \(\Delta_{\text{syn},k}\), a spike is emitted at step end if the post-update potential crosses threshold:

\[V_k^{\mathrm{post}} \ge V_{th}.\]

On spike:

\[V \leftarrow V_{reset}, \qquad r \leftarrow \left\lceil \frac{t_{ref}}{dt} \right\rceil,\]

where \(r\) is the integer refractory-step counter. While \(r > 0\), the membrane is clamped (no subthreshold integration is committed), then \(r\) decrements by one each simulation step.

3. Delta Synapses, Voltage Jumps, and Charge Interpretation

The change in membrane potential due to synaptic inputs can be formulated as:

\[\dot{\Delta}_{\text{syn}}(t) = \sum_{j} w_j \sum_k \delta(t-t_j^k-d_j) \;,\]

where \(j\) indexes either excitatory (\(w_j > 0\)) or inhibitory (\(w_j < 0\)) presynaptic neurons, \(k\) indexes the spike times of neuron \(j\), \(d_j\) is the delay from neuron \(j\), and \(\delta\) is the Dirac delta distribution. This implies that the jump in voltage upon a single synaptic input spike is

\[\Delta_{\text{syn}} = w \;,\]

where \(w\) is synaptic weight in mV. Positive weights are excitatory and negative weights are inhibitory.

The change in voltage caused by the synaptic input can be interpreted as being caused by individual post-synaptic currents (PSCs) given by

\[i_{\text{syn}}(t) = C_{\text{m}} \cdot w \cdot \delta(t) \;.\]

As a consequence, the total charge \(q\) transferred by a single PSC is

\[q = \int_0^{\infty} i_{\text{syn}}(t)\, dt = C_{\text{m}} \cdot w \;.\]

4. Assumptions, Constraints, and Computational Implications

• The model assumes unit-compatible parameters and broadcast-compatible shapes against self.varshape.

• V_min is optional; when provided, candidate voltage is clipped with max(V, V_min) before threshold evaluation.

• Per-step compute is \(O(\prod \mathrm{varshape})\) with vectorized elementwise operations.

• refractory_input=False discards delta events that arrive during refractory clamping, while refractory_input=True stores a decayed contribution that is released when refractoriness ends.

Note

This implementation uses exact integration for subthreshold dynamics and NEST-compatible conversion of refractory duration to grid steps via ceil(t_ref / dt).

Parameters:

• in_size (Size) – Population shape specification. All neuron parameters are broadcast to self.varshape derived from in_size.

• E_L (ArrayLike, optional) – Resting potential \(E_L\) in mV; scalar or array broadcastable to self.varshape. Default is -70. * u.mV.

• C_m (ArrayLike, optional) – Membrane capacitance \(C_m\) in pF; broadcastable to self.varshape. Expected positive for physical behavior. Default is 250. * u.pF.

• tau_m (ArrayLike, optional) – Membrane time constant \(\tau_m\) in ms; broadcastable to self.varshape. Expected positive. Default is 10. * u.ms.

• t_ref (ArrayLike, optional) – Absolute refractory duration \(t_{ref}\) in ms. Converted to integer simulation steps using ceil(t_ref / dt). Default is 2. * u.ms.

• V_th (ArrayLike, optional) – Spike threshold \(V_{th}\) in mV; broadcastable to self.varshape. Default is -55. * u.mV.

• V_reset (ArrayLike, optional) – Post-spike reset potential \(V_{reset}\) in mV; broadcastable to self.varshape. Default is -70. * u.mV.

• I_e (ArrayLike, optional) – Constant external current \(I_e\) in pA; scalar or array broadcastable to self.varshape. Default is 0. * u.pA.

• V_min (ArrayLike or None, optional) – Optional lower membrane bound \(V_{min}\) in mV. If None, no lower clipping is applied. Default is None.

• V_initializer (Callable, optional) – Initializer for membrane state V in init_state(). Output must be shape-compatible with self.varshape (and optional batch prefix). Default is braintools.init.Constant(-70. * u.mV).

• spk_fun (Callable, optional) – Surrogate spike function used by get_spike(). Default is braintools.surrogate.ReluGrad().

• spk_reset (str, optional) – Reset policy inherited from Neuron. 'hard' reproduces NEST hard reset behavior. Default is 'hard'.

• refractory_input (bool, optional) – If False, delta inputs during refractory are ignored. If True, refractory-arriving delta jumps are accumulated in refractory_spike_buffer and applied after refractory release. Default is False.

• ref_var (bool, optional) – If True, allocate boolean refractory state self.refractory for inspection. Default is False.

• name (str or None, optional) – Optional node name.

Parameter Mapping

Table 1 Parameter mapping to model symbols

Parameter	Type / shape / unit	Default	Math symbol	Semantics
in_size	Size; scalar/tuple	required	–	Defines population/state shape self.varshape.
E_L	ArrayLike, broadcastable to self.varshape (mV)	-70. * u.mV	\(E_L\)	Resting membrane potential.
C_m	ArrayLike, broadcastable (pF)	250. * u.pF	\(C_m\)	Membrane capacitance in subthreshold propagator.
tau_m	ArrayLike, broadcastable (ms)	10. * u.ms	\(\tau_m\)	Membrane leak time constant.
t_ref	ArrayLike, broadcastable (ms), step-converted by ceil	2. * u.ms	\(t_{ref}\)	Absolute refractory duration.
V_th and V_reset	ArrayLike, broadcastable (mV)	-55. * u.mV, -70. * u.mV	\(V_{th}\), \(V_{reset}\)	Threshold and reset voltages.
I_e	ArrayLike, broadcastable (pA)	0. * u.pA	\(I_e\)	Constant injected current.
V_min	ArrayLike broadcastable (mV) or None	None	\(V_{min}\)	Optional lower clamp before threshold test.
V_initializer	Callable	Constant(-70. * u.mV)	–	Initializes membrane state V.
spk_fun	Callable	ReluGrad()	–	Surrogate spike output function.
spk_reset	str	'hard'	–	Reset mode inherited from base neuron class.
refractory_input	bool	False	–	Controls refractory-period treatment of delta inputs.
ref_var	bool	False	–	Enables persistent refractory boolean state.
name	str | None	None	–	Optional node identifier.

Raises:

• ValueError – If parameter initialization or broadcasting fails (for example due to incompatible array shape in braintools.init.param).

• TypeError – If provided values are not compatible with expected units/types (mV, ms, pF, pA, or callable initializers/spike functions).

• KeyError – At runtime, if required simulation context entries (for example t or dt) are missing when update() is called.

• AttributeError – If update() is called before init_state() creates required state variables.

V

Membrane potential.

Type:

HiddenState

last_spike_time

Time of the last spike, used to implement the refractory period.

Type:

ShortTermState

refractory

Neuron refractory state (only present if ref_var=True).

Type:

HiddenState

Notes

• State variables are V, last_spike_time, refractory_step_count, and refractory_spike_buffer. refractory exists only when ref_var=True.

• Continuous current input x is combined with I_e through sum_current_inputs() in the same simulation step.

• Delta events from sum_delta_inputs() are interpreted in mV and added as instantaneous voltage jumps.

Examples

>>> import brainpy
>>> import brainstate
>>> import saiunit as u
>>> with brainstate.environ.context(dt=0.1 * u.ms):
...     neu = brainpy.state.iaf_psc_delta(in_size=10, t_ref=2.0 * u.ms)
...     neu.init_state()
...     with brainstate.environ.context(t=0.0 * u.ms):
...         spk = neu.update(x=500.0 * u.pA)
...     _ = spk.shape

>>> import brainpy
>>> import brainstate
>>> import saiunit as u
>>> with brainstate.environ.context(dt=0.1 * u.ms):
...     neu = brainpy.state.iaf_psc_delta(in_size=(2,), V_min=-80.0 * u.mV)
...     neu.init_state()
...     with brainstate.environ.context(t=0.0 * u.ms):
...         _ = neu.update(x=120.0 * u.pA)

References

[1]

Rotter S, Diesmann M (1999). Exact simulation of time-invariant linear systems with applications to neuronal modeling. Biological Cybernetics 81:381-402. DOI: https://doi.org/10.1007/s004220050570

[2]

Diesmann M, Gewaltig M-O, Rotter S, & Aertsen A (2001). State space analysis of synchronous spiking in cortical neural networks. Neurocomputing 38-40:565-571. DOI: https://doi.org/10.1016/S0925-2312(01)00409-X

See also

LIF

Leaky integrate-and-fire with current-based synapses

LIFRef

Leaky integrate-and-fire with refractory period (brainpy parameterization)

get_spike(V=None)

Evaluate surrogate spike activation for a voltage tensor.

Parameters:

V (ArrayLike or None, optional) – Membrane voltage input in mV, broadcast-compatible with self.varshape. If None, the method uses self.V.value.

Returns:

out – Surrogate spike output from self.spk_fun with the same shape as V (or self.V.value when V is None).

Return type:

dict

Raises:

TypeError – If V cannot participate in arithmetic with membrane parameters due to incompatible dtype/unit.

init_state(**kwargs)

Initialize membrane and refractory runtime states.

Parameters:

**kwargs – Unused compatibility parameters accepted by the base-state API.

Raises:

• ValueError – If initializer outputs cannot be broadcast to target state shape.

• TypeError – If initializer values are incompatible with required numeric/unit conversions.

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

Advance the neuron by one simulation step.

Parameters:

x (ArrayLike, optional) – External current input in pA for this step. It is combined with I_e and additional current sources from sum_current_inputs().

Returns:

out – Surrogate spike output from get_spike() with shape self.V.value.shape. The returned spike signal is computed from pre-reset post-threshold voltage v_post.

Return type:

jax.Array

Raises:

• KeyError – If simulation context does not provide required entries t or dt.

• AttributeError – If required states are missing because init_state() has not been called.

• TypeError – If input/state values are not unit-compatible with expected pA/mV arithmetic.