WilsonCowanSimplifiedStep

WilsonCowanSimplifiedStep#

class brainmass.WilsonCowanSimplifiedStep(in_size, tau_E=Quantity(1., 'ms'), a_E=1.2, theta_E=2.8, tau_I=Quantity(1., 'ms'), a_I=1.0, theta_I=4.0, w_exc=8.0, w_inh=12.0, 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 simplified connectivity.

This variant of the Wilson-Cowan model simplifies the connectivity by reducing the four connection weights to two parameters: one for excitatory connections (w_exc) and one for inhibitory connections (w_inh). This reduces the parameter space and can be useful for pedagogical purposes or initial exploration.

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.

  • a_E (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Excitatory gain (dimensionless). Broadcastable to in_size. Default is 1.2.

  • theta_E (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Excitatory threshold (dimensionless). Broadcastable to in_size. Default is 2.8.

  • 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.

  • a_I (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Inhibitory gain (dimensionless). Broadcastable to in_size. Default is 1..

  • theta_I (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Inhibitory threshold (dimensionless). Broadcastable to in_size. Default is 4.0.

  • w_exc (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Excitatory coupling strength (dimensionless). Applied to both E->E and E->I. Broadcastable to in_size. Default is 8..

  • w_inh (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Inhibitory coupling strength (dimensionless). Applied to both I->E and I->I. Broadcastable to in_size. Default is 12..

  • 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 simplified connectivity are

\[\tau_E \frac{dr_E}{dt} = -r_E(t) + \bigl[1 - r\, r_E(t)\bigr] F_E\bigl(w_{exc} r_E(t) - w_{inh} r_I(t) + I_E(t)\bigr),\]
\[\tau_I \frac{dr_I}{dt} = -r_I(t) + \bigl[1 - r\, r_I(t)\bigr] F_I\bigl(w_{exc} r_E(t) - w_{inh} r_I(t) + I_I(t)\bigr),\]

with the sigmoidal transfer function

\[F_j(x) = \frac{1}{1 + e^{-a_j (x - \theta_j)}} - \frac{1}{1 + e^{a_j \theta_j}},\quad j \in \{E, I\}.\]

Comparison to standard Wilson-Cowan:

  • Simplified connectivity: 2 weights (w_exc, w_inh) instead of 4 (wEE, wIE, wEI, wII)

  • Internal mapping: wEE = wIE = w_exc, wEI = wII = w_inh

  • Reduces parameter space from 11 to 8 parameters

  • Useful for pedagogical purposes and quick exploration

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.WilsonCowanSimplifiedStep(1)
>>> _ = model.init_all_states()
>>> out = model.update(rE_inp=0.5)
>>> out.shape
(1,)
__init__(in_size, tau_E=Quantity(1., 'ms'), a_E=1.2, theta_E=2.8, tau_I=Quantity(1., 'ms'), a_I=1.0, theta_I=4.0, w_exc=8.0, w_inh=12.0, 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