tvb.WilsonCowan

experimental.network_dynamics.dynamics.tvb.WilsonCowan(**kwargs)

Wilson-Cowan neural mass model with excitatory and inhibitory populations.

Two-population model representing the mean firing rates of excitatory (E) and inhibitory (I) neural populations with sigmoid activation functions and mutual interactions.

Notes

State equations:

\[ \begin{aligned} \frac{dE}{dt} &= \frac{-E + (k_e - r_e E) S_e}{\tau_e} \\ \frac{dI}{dt} &= \frac{-I + (k_i - r_i I) S_i}{\tau_i} \end{aligned} \]

Sigmoid activation functions:

\[ \begin{aligned} x_e &= \alpha_e (c_{ee} E - c_{ei} I + P - \theta_e + c_{\text{delayed}} + \text{lc}_e + \text{lc}_i) \\ x_i &= \alpha_i (c_{ie} E - c_{ii} I + Q - \theta_i + \text{lc}_e + \text{lc}_i) \end{aligned} \]

If shift_sigmoid=True (baseline-corrected):

\[S_e = c_e \left(\frac{1}{1+\exp(-a_e(x_e-b_e))} - \frac{1}{1+\exp(a_e b_e)}\right)\]

Otherwise (standard sigmoid):

\[S_e = \frac{c_e}{1 + \exp(-a_e(x_e - b_e))}\]

Attributes

Name	Type	Description
STATE_NAMES	tuple of str	State variables: ('E', 'I')
INITIAL_STATE	tuple of float	Default initial conditions: (0.1, 0.05)
AUXILIARY_NAMES	tuple of str	Auxiliary variables: ('S_e', 'S_i') (sigmoid outputs)
COUPLING_INPUTS	dict	Coupling specification: {'instant': 2, 'delayed': 1} Instant coupling has 2 components [E, I] feeding into both populations; delayed coupling feeds only into E population
DEFAULT_PARAMS	Bunch	Standard Wilson-Cowan parameters

References

Wilson & Cowan (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal, 12, 1-24.

Methods

Name	Description
dynamics	Compute Wilson-Cowan dynamics.

dynamics

experimental.network_dynamics.dynamics.tvb.WilsonCowan.dynamics(
    t,
    state,
    params,
    coupling,
    external,
)

Compute Wilson-Cowan dynamics.

Parameters

Name	Type	Description	Default
t	float	Current time (ms)	required
state	jnp.ndarray	Current state with shape [2, n_nodes] containing (E, I)	required
params	Bunch	Model parameters	required
coupling	Bunch	Coupling inputs with attributes: - .instant[0]: local E coupling - .instant[1]: local I coupling - .delayed[0]: long-range coupling	required
external	Bunch	External inputs (currently unused)	required

Returns

Name	Type	Description
derivatives	jnp.ndarray	State derivatives with shape [2, n_nodes]
auxiliaries	jnp.ndarray	Auxiliary variables with shape [2, n_nodes] containing (S_e, S_i) sigmoid outputs