gif_pop_psc_exp

class brainpy.state.gif_pop_psc_exp(in_size, N=100, tau_m=20.0, C_m=250.0, t_ref=4.0, lambda_0=10.0, Delta_V=2.0, E_L=0.0, V_reset=0.0, V_T_star=15.0, I_e=0.0, tau_syn_ex=3.0, tau_syn_in=6.0, tau_sfa=(300.0,), q_sfa=(0.5,), len_kernel=-1, BinoRand=True, rng_seed=0, name=None)

Population of generalized integrate-and-fire neurons (GIF) with exponential postsynaptic currents and adaptation.

gif_pop_psc_exp simulates a population of spike-response model neurons with multi-timescale adaptation and exponential postsynaptic currents. It directly models the population activity (sum of all spikes) without explicitly representing each individual neuron, following the algorithm of Schwalger et al. (2017) [1].

This is a brainpy.state re-implementation of the NEST simulator model of the same name, using NEST-standard parameterization.

1. Mathematical Model

The single neuron model is defined by the hazard function:

\[h(t) = \lambda_0 \exp\left(\frac{V_m(t) - E_{\mathrm{sfa}}(t)}{\Delta_V}\right)\]

After each spike, the membrane potential \(V_m\) is reset to \(V_{\mathrm{reset}}\). Spike frequency adaptation is implemented by a set of exponentially decaying traces, the sum of which is \(E_{\mathrm{sfa}}\). Upon a spike, each of the adaptation traces is incremented by the respective \(q_{\mathrm{sfa}}\) and decays with the respective time constant \(\tau_{\mathrm{sfa}}\).

2. Population Dynamics Algorithm

The model uses the algorithm from Figures 11 and 12 of [1] to simulate the population activity directly:

1. Membrane update: Exact exponential integration of the subthreshold membrane dynamics with exponential postsynaptic currents:

   \[C_m \frac{dV}{dt} = -\frac{C_m}{\tau_m}(V - E_L) + I_{\mathrm{syn,ex}} + I_{\mathrm{syn,in}} + I_e + I_{\mathrm{stim}}\]

2. Adaptation update: Multi-timescale adaptation state evolves as:

   \[E_{\mathrm{sfa}} = V_{T^*} + \sum_j Q_{30K,j} \cdot g_j\]

   where \(g_j\) tracks the convolution of past spike rates with the adaptation kernel.

3. Population escape rate: For each refractory cohort, compute the hazard-based escape (firing) probability. Free (non-refractory) neurons and neurons emerging from refractoriness are tracked separately.

4. Stochastic spike generation: The expected number of spikes is computed from the population escape rates, and the actual spike count is drawn from either a binomial or Poisson distribution.

5. History buffer rotation: Spike counts, survival counts, mean membrane potentials, and variances of each refractory cohort are maintained in rotating circular buffers.

3. Synaptic Currents

Exponential postsynaptic currents follow first-order dynamics:

\[\tau_{\mathrm{syn}} \frac{dy}{dt} = -y\]

Integrated exactly per time step:

\[y(t+h) = y_{\infty} + (y(t) - y_{\infty}) \cdot e^{-h/\tau_{\mathrm{syn}}}\]

The membrane contribution from synaptic currents is computed via the exact propagator for the coupled linear system of membrane and synaptic dynamics.

Connecting two population models corresponds to full connectivity of every neuron in each population. An approximation of random connectivity can be implemented by connecting populations using a bernoulli_synapse.

Parameters:

• in_size (int, sequence of int) – Shape of the population. Typically a scalar for 1D populations.

• N (int, optional) – Number of neurons represented by this population node. Default: 100.

• tau_m (float, optional) – Membrane time constant in milliseconds. Default: 20.0.

• C_m (float, optional) – Membrane capacitance in picofarads. Default: 250.0.

• t_ref (float, optional) – Duration of absolute refractory period in milliseconds. Default: 4.0.

• lambda_0 (float, optional) – Firing rate at threshold in 1/s (Hz). Default: 10.0.

• Delta_V (float, optional) – Noise level (voltage sensitivity) of the escape rate in millivolts. Determines how sharply the firing rate increases with membrane potential. Default: 2.0.

• E_L (float, optional) – Resting (leak) potential in millivolts. Default: 0.0.

• V_reset (float, optional) – Reset potential after spike in millivolts. Default: 0.0.

• V_T_star (float, optional) – Baseline threshold level in millivolts. Effective threshold is \(V_{T^*} + E_{\mathrm{sfa}}\). Default: 15.0.

• I_e (float, optional) – Constant external DC input current in picoamperes. Default: 0.0.

• tau_syn_ex (float, optional) – Excitatory synaptic time constant in milliseconds. Default: 3.0.

• tau_syn_in (float, optional) – Inhibitory synaptic time constant in milliseconds. Default: 6.0.

• tau_sfa (sequence of float, optional) – Adaptation time constants in milliseconds. Multiple timescales can be specified as a tuple. Default: (300.0,).

• q_sfa (sequence of float, optional) – Adaptation kernel amplitudes in millivolts. Must have the same length as tau_sfa. Default: (0.5,).

• len_kernel (int, optional) – History kernel length in time steps. If -1 (default), automatically computed to ensure convergence. Default: -1.

• BinoRand (bool, optional) – If True, use binomial distribution for spike generation. If False, use Poisson distribution. Default: True.

• rng_seed (int, optional) – Random number generator seed for reproducibility. Default: 0.

• name (str, optional) – Name of the population. Default: None.

Parameter Mapping

gif_pop_psc_exp	gif_psc_exp	relation
tau_m	g_L	tau_m = C_m / g_L
N	—	use N gif_psc_exp neurons

State Variables

The model exposes the following read-only properties:

State variable	Description
V_m	Mean membrane potential (mV)
I_syn_ex	Excitatory synaptic current (pA)
I_syn_in	Inhibitory synaptic current (pA)
n_spikes	Number of spikes in current step
n_expect	Expected number of spikes
theta_hat	Adaptive threshold (mV)
y0	External current input state (pA)

Notes

• As gif_pop_psc_exp represents many neurons in one node, it may generate many spikes per time step. The n_spikes state variable records the number of spikes emitted by the population each step.

• The computational cost of this model is largely independent of the number N of neurons represented.

• Defaults follow NEST C++ source for gif_pop_psc_exp.

• lambda_0 is specified in 1/s (as in NEST’s Python interface); the conversion factor of 0.001 to 1/ms is applied in the update equations exactly as in NEST (the factor 0.0005 in the trapezoidal escape rate).

• Synaptic spike weights are interpreted in current units; positive weights go to excitatory, negative to inhibitory channel.

• The model does not inherit from Neuron because it represents a population, not a single spiking neuron with a surrogate gradient.

• Parameters are stored as plain floats (unitless) to match NEST’s internal representation.

References

[1] (1,2,3,4)

Schwalger T, Deger M, Gerstner W (2017). Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size. PLoS Computational Biology, 13(4), e1005507. https://doi.org/10.1371/journal.pcbi.1005507

[2]

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

See also

gif_psc_exp, gif_cond_exp

Examples

Create a population of 100 neurons with default parameters:

>>> import brainpy.state as bp
>>> import saiunit as u
>>> pop = bp.gif_pop_psc_exp(1, N=100)
>>> pop.init_all_states()

Create a population with custom adaptation parameters:

>>> pop = bp.gif_pop_psc_exp(
...     1,
...     N=200,
...     tau_sfa=(100.0, 500.0),
...     q_sfa=(0.3, 0.6),
...     lambda_0=5.0
... )
>>> pop.init_all_states()

Simulate the population with external input:

>>> with brainstate.environ.context(dt=0.1 * u.ms):
...     pop.init_all_states()
...     for _ in range(1000):
...         n_spikes = pop.update(x=100.0)  # 100 pA input
...         print(f"Spikes: {n_spikes}, V_m: {pop.V_m:.2f}")

property I_syn_ex: float

Excitatory synaptic current in picoamperes.

Returns:

Current excitatory synaptic current (pA), evolving according to exponential dynamics with time constant tau_syn_ex.

Return type:

float

property I_syn_in: float

Inhibitory synaptic current in picoamperes.

Returns:

Current inhibitory synaptic current (pA), evolving according to exponential dynamics with time constant tau_syn_in.

Return type:

float

property V_m: float

Mean membrane potential in millivolts.

Returns:

Population-averaged membrane potential of all neurons (mV). Updated each time step by exact exponential integration.

Return type:

float

init_state(**kwargs)

Initialize all population state variables and history buffers.

Allocates and initializes circular buffers for tracking refractory cohorts, computes integration constants, and sets initial conditions following Procedure InitPopulations from Fig. 11 of [1].

Parameters:

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

Notes

• Computes integration time step h from the global brainstate dt.

• Determines history kernel length (automatic if len_kernel < 1).

• Initializes all neurons in the free (non-refractory) state.

• Precomputes adaptation kernel values and propagator constants.

• Resets the random number generator to rng_seed.

property n_expect: float

Expected number of spikes in the current time step.

Returns:

Mean spike count computed from the population escape rates before stochastic sampling. May exceed N in principle.

Return type:

float

property n_spikes: int

Number of spikes emitted in the current time step.

Returns:

Spike count drawn stochastically from the expected rate, ranging from 0 to N.

Return type:

int

reset_state(**kwargs)

Reset all population state variables to initial conditions.

Equivalent to calling init_state() again: re-initializes all history buffers, observable states, and the random number generator to their values immediately after construction.

Parameters:

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

property theta_hat: float

Adaptive threshold for non-refractory neurons in millivolts.

Returns:

Effective threshold including all adaptation contributions (mV). Equals V_T_star plus the sum of all adaptation traces.

Return type:

float

update(x=0.0)

Advance the population model by one time step.

Implements the full population update algorithm from Figures 11 and 12 of [1], including:

1. Exact exponential integration of membrane potential and synaptic currents

2. Multi-timescale adaptation state update

3. Escape rate computation for each refractory cohort

4. Stochastic spike generation (binomial or Poisson)

5. Circular buffer rotation for history tracking

Parameters:

x (float, optional) – External input current in picoamperes. Positive values are routed to the excitatory channel, negative values to the inhibitory channel. Default: 0.0.

Returns:

Number of spikes emitted by the population in this time step. Ranges from 0 to N. Stored in the n_spikes property.

Return type:

int

Notes

• Delta inputs (spike events) are collected from _delta_inputs and separated into excitatory and inhibitory components.

• Current inputs are collected from _current_inputs and accumulated into the y0 state for the next time step.

• The membrane potential V_m, synaptic currents I_syn_ex and I_syn_in, adaptive threshold theta_hat, and expected spike count n_expect are all updated and accessible via properties.

• The actual spike count is drawn stochastically from the expected count using either a binomial or Poisson distribution (controlled by BinoRand).

property y0: float

External current input state in picoamperes.

Returns:

Accumulated external current input (pA) that will be applied in the next time step.

Return type:

float