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Source code for tvb.simulator.coupling

# -*- coding: utf-8 -*-
#
#
#  TheVirtualBrain-Scientific Package. This package holds all simulators, and
# analysers necessary to run brain-simulations. You can use it stand alone or
# in conjunction with TheVirtualBrain-Framework Package. See content of the
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# (c) 2012-2020, Baycrest Centre for Geriatric Care ("Baycrest") and others
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# program.  If not, see <http://www.gnu.org/licenses/>.
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#
#   CITATION:
# When using The Virtual Brain for scientific publications, please cite it as follows:
#
#   Paula Sanz Leon, Stuart A. Knock, M. Marmaduke Woodman, Lia Domide,
#   Jochen Mersmann, Anthony R. McIntosh, Viktor Jirsa (2013)
#       The Virtual Brain: a simulator of primate brain network dynamics.
#   Frontiers in Neuroinformatics (7:10. doi: 10.3389/fninf.2013.00010)
#
#

"""
Coupling functions

The activity (state-variables) that have been propagated over the long-range
Connectivity pass through these functions before entering the equations
(Model.dfun()) describing the local dynamics.

The state-variable vector for the $k$-th node or region in the network can be expressed as:
Derivative = Noise + Local dynamics + Coupling(time delays).

More formally:

.. math::

         \\dot{\\Psi}_{k} = - \\Lambda\\left(\\Psi_{k}\\right) + Z \\left(\\Xi_{k} + \\sum_{j=1}^{l} u_{kj} \\Gamma_{v=2}[\\left(\\Psi_{k}(t),  \\Psi_{j}(t-\\tau_{kj}\\right)]\\right).


Here we compute the term Coupling(time delays) or 
:math:`\\sum_{j=1}^{l} u_{kj} \\Gamma_{v=2}[\\left(\\Psi_{k}(t),  \\Psi_{j}(t-\\tau_{kj}\\right)]`, 
where :math:`u_{kj}` are the elements of the weights matrix from a Connectivity datatype.

This term is equivalent to the dot product between the weights matrix (on the
left) and the delayed state vector. This order is important in the case 
case of an asymmetric connectivity matrix, where the
convention to distinguish target ($k$) and source ($j$) nodes is the
following:


.. math::

    \\left(\\begin{matrix} a & b \\
        c & d \end{matrix}\\right)

         C_{kj}  &= \left(\\begin{matrix} ^\mathrm{To}/_\mathrm{from} & 0 & 1 & 2 & \cdots & l \\
                                                           0         & 1  & 1  &  0 & 1  &  0 \\
                                                           1         & 1  & 1  &  0 & 1  &  0 \\
                                                           2         & 1  & 0  &  0 & 1  &  0 \\
                                                     \\vdots          & 1  & 0  &  1 & 0  &  1 \\
                                                           l         & 0  & 0  &  0 & 0  &  0 \\
                                                           \end{matrix}\\right)

.. NOTE: Our convention is the inverse of the BCT toolbox. Furthermore, this
         convention is consistent with the notation used in Physics and in our
         equation, ie, :math:`u_{kj}` matches row-column indexing and describes the
         connection strength from node j to node k


.. moduleauthor:: Stuart A. Knock <Stuart@tvb.invalid>
.. moduleauthor:: Noelia Montejo <Noelia@tvb.invalid>
.. moduleauthor:: Marmaduke Woodman <marmaduke.woodman@univ-amu.fr>
.. moduleauthor:: Paula Sanz Leon <Paula@tvb.invalid>

"""
import numpy
from tvb.basic.neotraits.api import HasTraits, NArray, Attr, Range
from .history import SparseHistory
from .common import simple_gen_astr


[docs]class Coupling(HasTraits): r""" The base class for Coupling functions. Instances of the Coupling class are called by the simulator in the following way: .. math:: k_i = coupling(g_ij, x_i, x_j) where g_ij is the connectivity weight matrix, x_i is the current state, x_j is the delayed state of the coupling variables chosen for the simulation, and k_i is the input to the ith node due to the coupling between the nodes. Coupling functions can all be defined as a combination of a pre-"synaptic" or pre-summation function, the summation over weighted afferents and the post-"synaptic" or post-summation function. Therefore, a Coupling subclass should not define the `__call__` method directly but rather appropriate `pre` and `post` methods, which are used by `Coupling.__call__` to compute the coupling correctly. Default implementations of `pre` and `post` are provided, which simply apply the connectivity to afferent activity, without scaling or other changes. .. automethod:: PreSigmoidal.__call__ """ def __call__(self, step, history): g_ij = history.es_weights x_i, x_j = history.query(step) x_i = x_i[numpy.newaxis].transpose((2, 1, 0, 3)) # (to, ncv, from, m) pre = self.pre(x_i, x_j) sum = (g_ij * pre).sum(axis=2) # (to, ncv, m) return self.post(sum).transpose((1, 0, 2)) # (ncv, to, m)
[docs] def pre(self, x_i, x_j): return x_j
[docs] def post(self, gx): return gx
[docs]class SparseCoupling(Coupling): """ A coupling implementation which takes advantage of a sparse weights structure to reduce the number of coupling terms evaluated. """ def _lri(self, nnz_row_el_idx): "Flat array of indices afferent, non-zero-weight connections." if not hasattr(self, '_cached_lri'): rows = numpy.r_[-1, nnz_row_el_idx] self._cached_lri, = numpy.argwhere(numpy.diff(rows)).T self._cached_nzr = numpy.unique(nnz_row_el_idx) self.log.debug('lri.size %d nzr.size %d', self._cached_lri.size, self._cached_nzr.size) return self._cached_lri, self._cached_nzr def __call__(self, step, history): h = history # type: SparseHistory x_i, x_j = h.query_sparse(step) assert x_i.shape == (h.n_cvar, h.n_node, h.n_mode) assert x_j.shape == (h.n_cvar, h.n_nnzw, h.n_mode) # ^ from (columns) sum = numpy.zeros_like(x_i) x_i = x_i[:, h.nnz_row_el_idx] assert x_i.shape == (h.n_cvar, h.n_nnzw, h.n_mode) # ^ to (rows) pre = self.pre(x_i, x_j) assert pre.shape == (h.n_cvar, h.n_nnzw, h.n_mode) weights_col = h.nnz_weights.reshape((h.n_nnzw, 1)) lri, nzr = self._lri(h.nnz_row_el_idx) sum[:, nzr] = numpy.add.reduceat(weights_col * pre, lri, axis=1) return self.post(sum)
[docs]class Linear(SparseCoupling): r""" Provides a linear coupling function of the following form .. math:: a x + b """ a = NArray( label=":math:`a`", default=numpy.array([0.00390625,]), domain=Range(lo=0.0, hi=1.0, step=0.01), doc="Rescales the connection strength while maintaining the ratio " "between different values.") b = NArray( label=":math:`b`", default=numpy.array([0.0]), doc="Shifts the base of the connection strength while maintaining " "the absolute difference between different values.")
[docs] def post(self, gx): return self.a * gx + self.b
def __str__(self): return simple_gen_astr(self, 'a b')
[docs]class Scaling(SparseCoupling): r""" Provides a simple scaling of the connectivity of the form .. math:: a x """ a = NArray( label="Scaling factor", default=numpy.array([0.00390625]), domain=Range(lo=0.0, hi=1.0, step=0.01), doc="Rescales the connection strength while maintaining " "the ratio between different values." )
[docs] def post(self, gx): return self.a * gx
def __str__(self): return simple_gen_astr(self, 'a')
[docs]class HyperbolicTangent(SparseCoupling): r""" Provides a sigmoidal coupling function of the form .. math:: a * (1 + tanh((x - midpoint)/sigma)) NB: This coupling function is applied pre-summation. For a post-summation sigmoidal, see `Sigmoidal`. """ a = NArray( label=":math:`a`", default=numpy.array([1.0]), domain=Range(lo=-1000.0, hi=1000.0, step=10.0), doc="Minimum of the sigmoid function") b = NArray( label=":math:`b`", default=numpy.array([1.0]), domain=Range(lo=-1.0, hi=1.0, step=10.0), doc="Scaling factor for the variable") midpoint = NArray( label="midpoint", default=numpy.array([0.0,]), domain=Range(lo=-1000.0, hi=1000.0, step=10.0), doc="Midpoint of the linear portion of the sigmoid") sigma = NArray( label=r":math:`\sigma`", default=numpy.array([1.0,]), domain=Range(lo=0.01, hi=1000.0, step=10.0), doc="Standard deviation of the coupling")
[docs] def pre(self, x_i, x_j): return self.a * (1 + numpy.tanh((self.b * x_j - self.midpoint) / self.sigma))
def __str__(self): return simple_gen_astr(self, 'a b midpoint sigma')
[docs]class Sigmoidal(Coupling): r""" Provides a sigmoidal coupling function of the form .. math:: c_{min} + (c_{max} - c_{min}) / (1.0 + \exp(-a(x-midpoint)/\sigma)) NB: using a = numpy.pi / numpy.sqrt(3.0) and the default parameter produces something close to the current default for Linear (a=0.00390625, b=0) over the linear portion of the sigmoid, with saturation at -1 and 1. """ cmin = NArray( label=":math:`c_{min}`", default=numpy.array([-1.0,]), domain=Range(lo=-1000.0, hi=1000.0, step=10.0), doc="""Minimum of the sigmoid function""",) cmax = NArray( label=":math:`c_{max}`", default=numpy.array([1.0,]), domain=Range(lo=-1000.0, hi=1000.0, step=10.0), doc="""Maximum of the sigmoid function""",) midpoint = NArray( label="midpoint", default=numpy.array([0.0,]), domain=Range(lo=-1000.0, hi=1000.0, step=10.0), doc="Midpoint of the linear portion of the sigmoid",) a = NArray( label=r":math:`a`", default=numpy.array([1.0,]), domain=Range(lo=0.01, hi=1000.0, step=10.0), doc="Scaling of sigmoidal",) sigma = NArray( label=r":math:`\sigma`", default=numpy.array([230.0,]), domain=Range(lo=0.01, hi=1000.0, step=10.0), doc="Standard deviation of the sigmoidal",) def __str__(self): return simple_gen_astr(self, 'cmin cmax midpoint a sigma')
[docs] def post(self, gx): return self.cmin + ((self.cmax - self.cmin) / (1.0 + numpy.exp(-self.a *((gx - self.midpoint) / self.sigma))))
[docs]class SigmoidalJansenRit(Coupling): r""" Provides a sigmoidal coupling function as described in the Jansen and Rit model, of the following form .. math:: c_{min} + (c_{max} - c_{min}) / (1.0 + \exp(-a(x-midpoint)/\sigma)) Assumes that x has have two state variables. """ cmin = NArray( label=":math:`c_{min}`", default=numpy.array([0.0,]), domain=Range(lo=-1000.0, hi=1000.0, step=10.0), doc="Minimum of the sigmoid function",) cmax = NArray( label=":math:`c_{max}`", default=numpy.array([2.0 * 0.0025,]), domain=Range(lo=-1000.0, hi=1000.0, step=10.0), doc="Maximum of the sigmoid function",) midpoint = NArray( label="midpoint", default=numpy.array([6.0,]), domain=Range(lo=-1000.0, hi=1000.0, step=10.0), doc="Midpoint of the linear portion of the sigmoid",) r = NArray( label=r":math:`r`", default=numpy.array([1.0,]), domain=Range(lo=0.01, hi=1000.0, step=10.0), doc="the steepness of the sigmoidal transformation",) a = NArray( label=r":math:`a`", default=numpy.array([0.56,]), domain=Range(lo=0.01, hi=1000.0, step=10.0), doc="Scaling of the coupling term",) def __str__(self): return simple_gen_astr(self, 'cmin cmax midpoint a r')
[docs] def pre(self, x_i, x_j): pre = self.cmax / (1.0 + numpy.exp(self.r * (self.midpoint - (x_j[:, 0] - x_j[:, 1])))) return pre[:, numpy.newaxis]
[docs] def post(self, gx): return self.a * gx
[docs]class PreSigmoidal(Coupling): r""" Provides a pre-summation sigmoidal coupling function with a static or dynamic and local or global threshold. .. math:: H * (Q + \tanh(G * (P*x - \theta))) The dynamic threshold as state variable given by the second state variable. With the coupling term, returns the direct node output for the dynamic threshold. """ H = NArray( label="H", default=numpy.array([0.5,]), domain=Range(lo=-100.0, hi=100.0, step=1.0), doc="Global Factor.",) Q = NArray( label="Q", default=numpy.array([1.,]), domain=Range(lo=-100.0, hi=100.0, step=1.0), doc="Average.",) G = NArray( label="G", default=numpy.array([60.,]), domain=Range(lo=-1000.0, hi=1000.0, step=1.), doc="Gain.",) P = NArray( label="P", default=numpy.array([1.,]), domain=Range(lo=-100.0, hi=100.0, step=0.01), doc="Excitation-Inhibition ratio.",) theta = NArray( label=":math:`\\theta`", default=numpy.array([0.5,]), domain=Range(lo=-100.0, hi=100.0, step=0.01), doc="Threshold.",) dynamic = Attr( field_type=bool, label="Dynamic", default=True, doc="Use dynamic threshold (otherwise static).",) globalT = Attr( field_type=bool, label=":math:`global_{\\theta}`", default=False, doc="Use global threshold (otherwise local).",) def __str__(self): return simple_gen_astr(self, 'H Q G P theta dynamic globalT')
[docs] def configure(self): """Set the right indirect call.""" super(PreSigmoidal, self).configure() self.sliceT = 0 if self.globalT else slice(None) # override __call__ directly simpler than pre/post form # TODO check use of arrays dims here
def __call__(self, step, history, na=numpy.newaxis): g_ij = history.es_weights x_i, x_j = history.query(step) if self.dynamic: _ = (self.P * x_j[:,0] - x_j[:,1,self.sliceT])[:,na] else: _ = self.P * x_j - self.theta[self.sliceT,na] A_j = self.H * (self.Q + numpy.tanh(self.G * _)) if self.dynamic: c_0 = (g_ij[:,0] * A_j[:,0]).sum(axis=0) c_1 = numpy.diag(A_j[:,0,:,0])[:, na] if self.globalT: c_1[:] = c_1.mean() return numpy.array([c_0, c_1]) else: # static threshold return (g_ij.transpose((2, 1, 0, 3)) * A_j).sum(axis=0)
[docs]class Difference(SparseCoupling): r""" Provides a difference coupling function, between pre and post synaptic activity of the form .. math:: a G_ij (x_j - x_i) """ a = NArray( label=":math:`a`", default=numpy.array([0.1,]), domain=Range(lo=0.0, hi=10., step=0.1), doc="Rescales the connection strength.",) def __str__(self): return simple_gen_astr(self, 'a')
[docs] def pre(self, x_i, x_j): return x_j - x_i
[docs] def post(self, gx): return self.a * gx
[docs]class Kuramoto(SparseCoupling): r""" Provides a Kuramoto-style coupling, a periodic difference of the form .. math:: a / N G_ij sin(x_j - x_i) """ a = NArray( label=":math:`a`", default=numpy.array([1.0,]), domain=Range(lo=0.0, hi=1.0, step=0.01), doc="Rescales the connection strength.",) def __str__(self): return simple_gen_astr(self, 'a')
[docs] def pre(self, x_i, x_j): return numpy.sin(x_j - x_i)
[docs] def post(self, gx): return self.a / gx.shape[0] * gx