WilsonCowanLinearStep

WilsonCowanLinearStep#

class brainmass.WilsonCowanLinearStep(in_size, tau_E=Quantity(1., 'ms'), tau_I=Quantity(1., 'ms'), wEE=0.8, wIE=0.3, wEI=1.0, wII=0.85, r=1.0, noise_E=None, noise_I=None, rE_init=Constant(value=0.0), rI_init=Constant(value=0.0), method='exp_euler')#

Wilson-Cowan neural mass model with linear (ReLU) transfer function.

This variant of the Wilson-Cowan model replaces the sigmoidal transfer function with a rectified linear unit (ReLU) function: [x]+ = max(0, x). This removes the need for sigmoid gain and threshold parameters, simplifies the computational graph, and can be more gradient-friendly for optimization tasks.

Parameters:

  • in_size (int | Sequence[int] | integer | Sequence[integer]) – Spatial shape of each population (E and I). Can be an int, a tuple of ints, or any size compatible with brainstate.

  • tau_E (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Excitatory time constant with unit of time (e.g., 1. * u.ms). Broadcastable to in_size. Default is 1. * u.ms.

  • tau_I (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Inhibitory time constant with unit of time (e.g., 1. * u.ms). Broadcastable to in_size. Default is 1. * u.ms.

  • wEE (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – E->E coupling strength (dimensionless). Broadcastable to in_size. Default is 0.8.

  • wIE (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – E->I coupling strength (dimensionless). Broadcastable to in_size. Default is 0.3.

  • wEI (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – I->E coupling strength (dimensionless). Broadcastable to in_size. Default is 1.0.

  • wII (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – I->I coupling strength (dimensionless). Broadcastable to in_size. Default is 0.85.

  • r (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Refractory parameter (dimensionless) that limits maximum activation. Broadcastable to in_size. Default is 1..

  • noise_E (Noise) – Additive noise process for the excitatory population. If provided, its output is added to rE_inp at each update. Default is None.

  • noise_I (Noise) – Additive noise process for the inhibitory population. If provided, its output is added to rI_inp at each update. Default is None.

  • rE_init (Callable) – Parameter for the excitatory state rE. Default is braintools.init.Constant(0.0).

  • rI_init (Callable) – Parameter for the inhibitory state rI. Default is braintools.init.Constant(0.0).

  • method (str) – The numerical integration method to use. One of 'exp_euler', 'euler', 'rk2', or 'rk4', that is implemented in braintools.quad. Default is 'exp_euler'.

Return type:

Any

rE#

Excitatory population activity (dimensionless). Shape equals (batch?,) + in_size after init_state.

Type:

brainstate.HiddenState

rI#

Inhibitory population activity (dimensionless). Shape equals (batch?,) + in_size after init_state.

Type:

brainstate.HiddenState

Notes

The continuous-time Wilson-Cowan equations with ReLU transfer are

\[\tau_E \frac{dr_E}{dt} = -r_E(t) + \bigl[1 - r\, r_E(t)\bigr] \bigl[w_{EE} r_E(t) - w_{EI} r_I(t) + I_E(t)\bigr]_+,\]
\[\tau_I \frac{dr_I}{dt} = -r_I(t) + \bigl[1 - r\, r_I(t)\bigr] \bigl[w_{IE} r_E(t) - w_{II} r_I(t) + I_I(t)\bigr]_+,\]

where \([x]_+ = \max(0, x)\) is the rectified linear unit.

Comparison to standard Wilson-Cowan:

  • ReLU transfer function instead of sigmoid

  • Removed sigmoid parameters: a_E, a_I, theta_E, theta_I

  • Reduces parameter space from 11 to 7 parameters

  • Simpler computational graph, faster evaluation

  • More gradient-friendly for optimization

  • Important: Default weights are scaled down by ~13-15x for stability

References

Wilson, H. R., & Cowan, J. D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal, 12, 1–24.

Examples

>>> import brainmass
>>> import brainstate
>>> import brainunit as u
>>> brainstate.environ.set(dt=0.1 * u.ms)
>>> model = brainmass.WilsonCowanLinearStep(1)
>>> _ = model.init_all_states()
>>> out = model.update(rE_inp=0.5)
>>> out.shape
(1,)
__init__(in_size, tau_E=Quantity(1., 'ms'), tau_I=Quantity(1., 'ms'), wEE=0.8, wIE=0.3, wEI=1.0, wII=0.85, r=1.0, noise_E=None, noise_I=None, rE_init=Constant(value=0.0), rI_init=Constant(value=0.0), method='exp_euler')[source]#
Parameters:
drE(rE, rI, ext)[source]#

Right-hand side for the excitatory population.

Must be implemented by subclasses.

Parameters:

  • rE (array-like) – Excitatory activity (dimensionless).

  • rI (array-like) – Inhibitory activity (dimensionless), broadcastable to rE.

  • ext (array-like or scalar) – External input to E.

Returns:

Time derivative drE/dt with unit of 1/time.

Return type:

array-like

drI(rI, rE, ext)[source]#

Right-hand side for the inhibitory population.

Must be implemented by subclasses.

Parameters:

  • rI (array-like) – Inhibitory activity (dimensionless).

  • rE (array-like) – Excitatory activity (dimensionless), broadcastable to rI.

  • ext (array-like or scalar) – External input to I.

Returns:

Time derivative drI/dt with unit of 1/time.

Return type:

array-like