LinearStep

class brainmass.LinearStep(in_size, gamma=-10.0, init_x=Constant(value=0.01), noise_x=None, method='exp_euler')

Linear neural-mass node with damping (TVB Linear model).

A single-state linear model with a damping coefficient, used in The Virtual Brain as a canonical baseline for validating simulation pipelines and network coupling without nonlinear complications [1]. Despite its simplicity it is a genuine network node: the long-range and local coupling enter additively.

\[\frac{dx}{dt} = \gamma\,x + c,\]

where \(x\) is the (dimensionless) node activity, \(\gamma\) the damping coefficient, and \(c\) the summed coupling/external input. For a stable node \(\gamma\) must be negative and its magnitude should exceed the node’s in-degree; with \(c = 0\) the activity relaxes exponentially, \(x(t) = x_0 e^{\gamma t}\).

This is distinct from ThresholdLinearStep, which is a two-population (E/I) threshold-linear rate model.

Parameters:

• in_size (int | Sequence[int] | integer | Sequence[integer]) – Spatial shape of the node. An int or tuple of int; gamma broadcasts to this shape.

• gamma (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param) – Damping coefficient (dimensionless). Must be negative for stability. Default is -10.0.

• init_x (Callable) – Initializer for the activity state x. Default is braintools.init.Constant(0.01).

• noise_x (Noise) – Additive noise process. If provided, its output is added to the input x_inp at each update. Default is None.

• method (str) – Integration method, 'exp_euler' (default) or any braintools.quad method (e.g. 'rk4'). The exponential-Euler step is exact for this linear system.

Return type:

Any

x

Node activity (dimensionless). Shape (batch?,) + in_size.

Type:

brainstate.HiddenState

Notes

The state is dimensionless and dx() carries unit 1/ms, so an exponential-Euler step with dt in milliseconds is consistent (the convention shared by the other *Step models in this package).

References

[1]

P. Sanz-Leon, S. A. Knock, A. Spiegler, V. K. Jirsa (2015). Mathematical framework for large-scale brain network modeling in The Virtual Brain. NeuroImage, 111, 385-430. https://doi.org/10.1016/j.neuroimage.2015.01.002

Examples

>>> import brainmass
>>> import brainstate
>>> import brainunit as u
>>> model = brainmass.LinearStep(in_size=1, gamma=-5.0)
>>> _ = brainstate.nn.init_all_states(model)
>>> with brainstate.environ.context(dt=0.1 * u.ms):
...     x = model.update()
>>> x.shape
(1,)

__init__(in_size, gamma=-10.0, init_x=Constant(value=0.01), noise_x=None, method='exp_euler')

Parameters:

• in_size (int | Sequence[int] | integer | Sequence[integer])

• gamma (Callable | Array | ndarray | bool | number | bool | int | float | complex | Quantity | Param)

• init_x (Callable)

• noise_x (Noise)

• method (str)

derivative(state, t, x_inp)

dx(x, x_inp)

Right-hand side for the activity x.

Parameters:

• x (array-like) – Current activity (dimensionless).

• x_inp (array-like or scalar) – Summed coupling/external input (includes noise if enabled).

Returns:

Time derivative dx/dt with unit 1/ms.

Return type:

array-like

init_state(batch_size=None, **kwargs)

Allocate the activity state x.

Parameters:

batch_size (int or None, optional) – Optional leading batch dimension. If None, no batch dimension is used. Default is None.

update(x_inp=None)

Advance the node by one time step.

Parameters:

x_inp (array-like or scalar or None, optional) – Summed coupling/external input. If None, treated as zero. If noise_x is set, its output is added. Default is None.

Returns:

The updated activity x, same shape as the internal state.

Return type:

array-like