iaf_psc_alpha_ps

class brainpy.state.iaf_psc_alpha_ps(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"), tau_syn_ex=Quantity(2., "ms"), tau_syn_in=Quantity(2., "ms"), 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', ref_var=False, name=None)

NEST-compatible iaf_psc_alpha_ps with precise spike timing.

Description

iaf_psc_alpha_ps is a current-based leaky integrate-and-fire neuron with alpha-shaped excitatory/inhibitory postsynaptic currents (PSCs), fixed absolute refractoriness, and off-grid spike/event timing. The implementation matches NEST models/iaf_psc_alpha_ps.{h,cpp} semantics: event-driven mini-step splitting inside each global dt interval, exact linear propagators for alpha states, and bisection-based sub-step threshold-time localization.

1. Continuous-Time Model and Alpha Current State-Space

Define \(U = V_m - E_L\) and \(I_\mathrm{syn}=I_\mathrm{ex}+I_\mathrm{in}\). Subthreshold dynamics are

\[\frac{dU}{dt} = -\frac{U}{\tau_m} + \frac{I_\mathrm{syn} + I_e + y_\mathrm{in}}{C_m}.\]

For each channel \(X\in\{\mathrm{ex},\mathrm{in}\}\), alpha PSCs use a two-state system:

\[\frac{d\,dI_X}{dt} = -\frac{dI_X}{\tau_{\mathrm{syn},X}}, \qquad \frac{dI_X}{dt} = dI_X - \frac{I_X}{\tau_{\mathrm{syn},X}}.\]

This realizes normalized kernel

\[i_X(t) = \frac{e}{\tau_{\mathrm{syn},X}} t e^{-t/\tau_{\mathrm{syn},X}} \Theta(t),\]

so a spike weight \(w\) (pA) is injected into derivative states as \(dI_\mathrm{ex}\leftarrow dI_\mathrm{ex}+\frac{e}{\tau_{\mathrm{syn,ex}}}w\) for \(w\ge 0\) and \(dI_\mathrm{in}\leftarrow dI_\mathrm{in}+\frac{e}{\tau_{\mathrm{syn,in}}}w\) for \(w<0\) (inhibitory channel stays negative by sign convention).

2. Exact Mini-Step Propagation and Precise Threshold Crossing

For each local interval \(\Delta t\) between two ordered event offsets, the code uses exact closed-form updates:

\[dI_X(t+\Delta t) = e^{-\Delta t/\tau_{\mathrm{syn},X}} dI_X(t),\]

\[I_X(t+\Delta t) = e^{-\Delta t/\tau_{\mathrm{syn},X}} \big(I_X(t) + \Delta t\, dI_X(t)\big),\]

\[U(t+\Delta t) = U(t) + \left(e^{-\Delta t/\tau_m}-1\right)U(t) + P_{30}(I_e+y_\mathrm{in}) + \sum_X \left(P_{31,X} dI_X(t) + P_{32,X} I_X(t)\right),\]

with \(P_{30}=\tau_m(1-e^{-\Delta t/\tau_m})/C_m\) and \(P_{31,X}, P_{32,X}\) evaluated by iaf_psc_alpha._alpha_propagator_p31_p32() (including stable handling near \(\tau_m\approx\tau_{\mathrm{syn},X}\)).

If \(U\) crosses \(U_{th}=V_{th}-E_L\) inside a mini-step, the crossing time solves \(f(\delta)=U(\delta)-U_{th}=0\) using bounded bisection (64 iterations), producing off-grid spike offset spike_off = dt - (local_time + delta).

3. Event Ordering, Refractory Pseudo-Event, and Timing Convention

Off-grid events are sorted by offset in descending order, where offset is measured from the right boundary of the current step (\(0\) at step end, \(dt\) at step start). Each neuron can also insert a refractory-release pseudo-event at stored last_spike_offset when step_idx + 1 - last_spike_step == ceil(t_ref / dt).

On spike emission:

• membrane state is reset to V_reset,

• refractory flag is set,

• last_spike_step, last_spike_offset, last_spike_time are updated with precise sub-step timing.

4. Assumptions, Constraints, and Computational Implications

• Construction constraints enforce C_m > 0, tau_m > 0, tau_syn_ex > 0, tau_syn_in > 0, and V_reset < V_th.

• If V_min is set, V_reset >= V_min is required.

• Runtime requires ceil(t_ref / dt) >= 1; otherwise update fails.

• x is ring-buffered current input: values supplied at step n are consumed as y_input in step n+1.

• Update is vectorized over self.varshape using array operations. With K within-step events, cost is \(O(|\mathrm{varshape}| \cdot K)\), plus root-search work when threshold is crossed.

Parameters:

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

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

• C_m (ArrayLike, optional) – Membrane capacitance \(C_m\) in pF. Must be strictly positive after broadcasting to self.varshape. Default is 250. * u.pF.

• tau_m (ArrayLike, optional) – Membrane time constant \(\tau_m\) in ms. Must be strictly positive. Default is 10. * u.ms.

• t_ref (ArrayLike, optional) – Absolute refractory time \(t_{ref}\) in ms. Converted at runtime to grid steps via ceil(t_ref / dt). Must yield at least one step. 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. Must satisfy V_reset < V_th elementwise. Default is -70. * u.mV.

• tau_syn_ex (ArrayLike, optional) – Excitatory alpha time constant \(\tau_{\mathrm{syn,ex}}\) in ms. Strictly positive. Default is 2. * u.ms.

• tau_syn_in (ArrayLike, optional) – Inhibitory alpha time constant \(\tau_{\mathrm{syn,in}}\) in ms. Strictly positive. Default is 2. * u.ms.

• I_e (ArrayLike, optional) – Constant external current \(I_e\) in pA, broadcastable to self.varshape. Added in each mini-step membrane update. Default is 0. * u.pA.

• V_min (ArrayLike or None, optional) – Optional lower voltage clamp \(V_{min}\) in mV. When provided, membrane candidates are clipped by max(V, V_min) before threshold tests. None disables clipping. Default is None.

• V_initializer (Callable, optional) – Initializer for membrane state V used in init_state(). Must return values unit-compatible with mV. Default is braintools.init.Constant(-70. * u.mV).

• spk_fun (Callable, optional) – Surrogate spike function used by get_spike() and returned by update(). It receives normalized threshold distance and returns a spike-like array broadcastable to neuron shape. Default is braintools.surrogate.ReluGrad().

• spk_reset (str, optional) – Reset policy passed to Neuron. 'hard' matches NEST hard-reset behavior. Default is 'hard'.

• ref_var (bool, optional) – If True, creates exposed state self.refractory mirroring self.is_refractory for inspection. Default is False.

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

Parameter Mapping

Table 5 Parameter mapping to model symbols

Parameter	Type / shape / unit	Default	Math symbol	Semantics
in_size	Size; scalar/tuple	required	–	Defines population shape self.varshape.
E_L	ArrayLike, broadcastable to self.varshape (mV)	-70. * u.mV	\(E_L\)	Resting potential; membrane offset origin.
C_m	ArrayLike, broadcastable (pF), > 0	250. * u.pF	\(C_m\)	Membrane capacitance in all propagators.
tau_m	ArrayLike, broadcastable (ms), > 0	10. * u.ms	\(\tau_m\)	Membrane leak time constant.
t_ref	ArrayLike, broadcastable (ms), runtime ceil(t_ref/dt) >= 1	2. * u.ms	\(t_{ref}\)	Absolute refractory duration.
V_th and V_reset	ArrayLike, broadcastable (mV), with V_reset < V_th	-55. * u.mV, -70. * u.mV	\(V_{th}\), \(V_{reset}\)	Threshold and reset levels.
tau_syn_ex and tau_syn_in	ArrayLike, broadcastable (ms), each > 0	2. * u.ms	\(\tau_{\mathrm{syn,ex}}\), \(\tau_{\mathrm{syn,in}}\)	Alpha PSC decay constants.
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 membrane bound.
V_initializer	Callable returning mV-compatible values	Constant(-70. * u.mV)	–	Initial membrane state initializer.
spk_fun	Callable	ReluGrad()	–	Surrogate spike output nonlinearity.
spk_reset	str	'hard'	–	Reset mode inherited from base Neuron.
ref_var	bool	False	–	Allocate exposed refractory mirror state.
name	str | None	None	–	Optional node name.

Raises:

• ValueError – If parameter constraints are violated (for example C_m <= 0, tau_m <= 0, tau_syn_ex <= 0, tau_syn_in <= 0, V_reset >= V_th, V_reset < V_min), if refractory duration in steps is below one, or if any spike_events offset is outside [0, dt].

• TypeError – If supplied quantities are not unit-compatible with expected units (mV, ms, pA, pF) during conversion.

• KeyError – If simulation context keys such as t or dt are missing when update() is called.

• AttributeError – If update() is called before init_state() creates required states (for example V or synaptic buffers).

Notes

• spike_events accepts (offset, weight) tuples or {'offset': ..., 'weight': ...} dicts. Offsets are in ms and measured from the right step boundary (NEST convention).

• Positive event weights update the excitatory derivative state; negative event weights update inhibitory derivative state.

• The implementation computes all internal propagators in float64 NumPy space and writes back BrainUnit states afterward.

• last_spike_time stores precise absolute spike time in ms and is stop-gradient wrapped.

Examples

>>> import brainpy
>>> import brainstate
>>> import saiunit as u
>>> with brainstate.environ.context(dt=0.1 * u.ms):
...     neu = brainpy.state.iaf_psc_alpha_ps(in_size=(2,), I_e=220.0 * u.pA)
...     neu.init_state()
...     with brainstate.environ.context(t=1.0 * u.ms):
...         spk = neu.update()
...     _ = spk.shape

>>> import brainpy
>>> import brainstate
>>> import saiunit as u
>>> with brainstate.environ.context(dt=0.1 * u.ms):
...     neu = brainpy.state.iaf_psc_alpha_ps(in_size=1)
...     neu.init_state()
...     ev = [{'offset': 0.08 * u.ms, 'weight': 120.0 * u.pA}]
...     with brainstate.environ.context(t=0.0 * u.ms):
...         _ = neu.update(spike_events=ev)

References

[1]

NEST source: models/iaf_psc_alpha_ps.h and models/iaf_psc_alpha_ps.cpp.

[2]

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

[3]

Morrison A, Straube S, Plesser HE, Diesmann M (2007). Exact subthreshold integration with continuous spike times in discrete time neural network simulations. Neural Computation 19(1):47-79. DOI: https://doi.org/10.1162/neco.2007.19.1.47

[4]

Hanuschkin A, Kunkel S, Helias M, Morrison A, Diesmann M (2010). A general and efficient method for incorporating exact spike times in globally time-driven simulations. Frontiers in Neuroinformatics. DOI: https://doi.org/10.3389/fninf.2010.00113

get_spike(V=None)

Evaluate surrogate spike output from membrane voltage.

Parameters:

V (ArrayLike, optional) – Voltage values with shape broadcastable to self.varshape and units compatible with mV. If None, uses current state self.V.value.

Returns:

Surrogate spike activation produced by spk_fun((V - V_th) / (V_th - V_reset)).

Return type:

ArrayLike

init_state(**kwargs)

Initialize persistent and short-term state variables.

Parameters:

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

Raises:

• ValueError – If an initializer cannot be broadcast to requested shape.

• TypeError – If initializer outputs have incompatible units/dtypes for the corresponding state variables.

update(x=Quantity(0., 'pA'), spike_events=None)

Advance one simulation step with optional precise within-step events.

Parameters:

• x (ArrayLike, optional) – Continuous external current in pA for the current global step. Value is accumulated through sum_current_inputs() and written to self.y_input for use in the next step (one-step buffering). Scalar or array-like broadcastable to self.V.value.shape. Default is 0. * u.pA.

• spike_events (Iterable[tuple[Any, Any] | dict[str, Any]] or None, optional) – Optional off-grid spike events within this dt step. Each item is either (offset, weight) or {'offset': ..., 'weight': ...}, where offset is in ms from the right step edge and weight is in pA. offset must satisfy 0 <= offset <= dt. Positive weights target excitatory alpha derivative state; negative weights target inhibitory alpha derivative state. None means no extra within-step events. On-grid delta inputs collected from sum_delta_inputs() are still included at offset=0.

Returns:

out – Spike output from get_spike() with shape self.V.value.shape. Values are surrogate spikes from self.spk_fun evaluated on threshold-scaled membrane potential after precise-time integration and event handling.

Return type:

jax.Array

Raises:

• ValueError – If computed refractory steps satisfy ceil(t_ref / dt) < 1 or if any event offset is outside [0, dt].

• KeyError – If simulation context values t or dt are missing.

• TypeError – If provided quantities are not unit-compatible with ms/pA during conversion of x or spike_events.

• AttributeError – If called before required states are initialized via init_state().