WilsonCowanStep

WilsonCowanStep#

class brainmass.WilsonCowanStep(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, wEE=12.0, wIE=4.0, wEI=13.0, wII=11.0, r=1.0, noise_E=None, noise_I=None, rE_init=ZeroInit(unit=1), rI_init=ZeroInit(unit=1), method='exp_euler')#

Wilson–Cowan neural mass model.

The model captures the interaction between an excitatory (E) and an inhibitory (I) neural population. It is widely used to study neural oscillations, multistability, and other emergent dynamics in cortical circuits.

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.

  • wEE (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – E→E coupling strength (dimensionless). Broadcastable to in_size. Default is 12..

  • wIE (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – E→I coupling strength (dimensionless). Broadcastable to in_size. Default is 4..

  • wEI (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – I→E coupling strength (dimensionless). Broadcastable to in_size. Default is 13..

  • wII (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – I→I coupling strength (dimensionless). Broadcastable to in_size. Default is 11..

  • 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.ZeroInit().

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

  • 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 are

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

with the sigmoidal transfer function

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

References

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

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