WilsonCowanSymmetricStep

WilsonCowanSymmetricStep#

class brainmass.WilsonCowanSymmetricStep(in_size, tau=Quantity(1., 'ms'), a=1.1, theta=3.4, wEE=12.0, wIE=4.0, wEI=13.0, wII=11.0, 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 symmetric parameters.

This variant of the Wilson-Cowan model uses symmetric parameters for the excitatory and inhibitory populations, i.e., both populations share the same time constant \(tau\), gain \(a\), and threshold \(theta\). This reduces the parameter space and can be useful for fitting or when assuming similar dynamics for both populations.

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

  • a (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Shared gain (dimensionless). Broadcastable to in_size. Default is 1.1.

  • theta (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Shared threshold (dimensionless). Broadcastable to in_size. Default is 3.4.

  • 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.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 symmetric parameters are

\[\tau \frac{dr_E}{dt} = -r_E(t) + \bigl[1 - r\, r_E(t)\bigr] F\bigl(w_{EE} r_E(t) - w_{EI} r_I(t) + I_E(t); a, \theta\bigr),\]
\[\tau \frac{dr_I}{dt} = -r_I(t) + \bigl[1 - r\, r_I(t)\bigr] F\bigl(w_{IE} r_E(t) - w_{II} r_I(t) + I_I(t); a, \theta\bigr),\]

with the sigmoidal transfer function

\[F(x; a, \theta) = \frac{1}{1 + e^{-a (x - \theta)}} - \frac{1}{1 + e^{a \theta}}.\]

Comparison to standard Wilson-Cowan:

  • Unified parameters: \(\tau\) instead of \(\tau_E, \tau_I\)

  • Unified sigmoid: \(a\) instead of \(a_E, a_I\)

  • Unified threshold: \(\theta\) instead of \(\theta_E, \theta_I\)

  • Reduces parameter space from 11 to 6 parameters

  • Useful for fitting data when E/I symmetry is assumed

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.WilsonCowanSymmetricStep(1)
>>> _ = model.init_all_states()
>>> out = model.update(rE_inp=0.5)
>>> out.shape
(1,)
__init__(in_size, tau=Quantity(1., 'ms'), a=1.1, theta=3.4, wEE=12.0, wIE=4.0, wEI=13.0, wII=11.0, 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