1.4. Edge Definitions

In this tutorial, we will go through the different options you have to define edge templates, i.e. formulations of the dynamics of signal propagation along the edges in a network. As an example, we will use a model of \(N = 2\) interconnected leaky integrator units, the evolution equations of which are given by

\[\dot r_i = - \frac{(r_0 - r_i)}{\tau} + u(t) + \sum_{j=1}^N J_{ij} f(r_j),\]

with individual rates \(r_i\), a global time scale \(\tau\), extrinsic input \(u(t)\), coupling weights \(J_{ij}\) and a coupling function \(f\). The latter represents the signal transformation that can happen during signal propagation along an edge Below, we will go through the options you have to specify \(f\) in PyRates.

As preparation, lets import all required packages and load the leaky integrator model into PyRates.

# external imports
import numpy as np
import matplotlib.pyplot as plt

# pyrates imports
from pyrates import CircuitTemplate, NodeTemplate, EdgeTemplate, OperatorTemplate, clear

# node definition
li = NodeTemplate.from_yaml("model_templates.base_templates.li_node")

The node template li includes the leaky-integrator equation, but we have not specified \(f\) yet. For this, we can use an EdgeTemplate. Below, we will go through several choices of edge templates.

1.4.1. Example 1: A simple instantaneous transform

We start with a simple instantaneous transform, i.e. a function \(f\) that does not depend on time. Aas an example, we will use the hyperbolic tangent as our coupling function.

tanh_op = OperatorTemplate(name="tanh_op", equations="m = tanh(x)",
                           variables={"m": "output", "x": "input"})
tanh_edge = EdgeTemplate(name="tanh_edge", operators=[tanh_op])

In the above code line, we first created an operator template that implements the hyperbolic tangent. We then used this operator template as the single operator from which to create an edge template, thus defining \(f = tanh\) as our coupling function. Now, we can use this edge template to define a circuit:

tanh_net = CircuitTemplate(name="tanh_net", nodes={"li1": li, "li2": li},
                           edges=[("li1/li_op/r", "li2/li_op/m_in", tanh_edge, {"weight": 5.0})])

This circuit contains a single edge from node li1 to node li2, which uses the tanh_edge as its coupling function. To make sure the edge works, lets perform a numerical simulation, where we ramp up the extrinsic input \(u(t)\) to li1. We would expect (1) the input to continuously increase \(r_1\), (2) that increases in \(r_1\) lead to an increase in \(r_2\) due to the coupling, and (3) that increases in \(r_2\) will eventually hit a ceiling due to the hyperbolic tangent being the coupling function. Let’s see if our expectations are met:

T = 100.0
dt = 1e-2
inp = np.linspace(0.0, 10.0, int(T/dt))

res = tanh_net.run(T, dt, inputs={"li1/li_op/u": inp}, outputs={"r1": "li1/li_op/r", "r2": "li2/li_op/r"},
                   solver="scipy", method="RK23")


As we can see, our expectations were all met.

1.4.2. Example 2: A kernel convolution

Even though we used a very simple coupling function above, edge templates can be made just as complex as node templates. In this example, we will use a time-dependent coupling function \(f(r, t)\). A typical example for signal propagation in biophysical systems is the convolution of the source signal of an edge with an impulse response function (or response kernel). Here, we choose a convolution with an alpha kernel by defining

\[f(r, t) = \int_0^{t} = \frac{t-t'}{\tau_{\alpha}^2} \exp(\frac{t-t'}{\tau_{\alpha}}) r(t') dt',\]

with alpha kernel time constant \(\tau_{\alpha}\). This convolution integral can be solved analytically, yielding the following set of coupled differential equations:

\[\begin{split}\tau_{\alpha} \dot x = y, \\ \tau_{\alpha} \dot y = -2y - x + \tau_{\alpha} r.\end{split}\]

Thus, we can implement the alpha kernel convolution via an operator governed by these two differential equations:

# set up alpha convolution operator
alpha_op = OperatorTemplate(name="alpha_op",
                            equations=["x' = z/tau",
                                       "z' = r_in - (2*z + x)/tau"],
                            variables={"x": "output", "z": "variable", "tau": 10.0, "r_in": "input"})

# set up edge with alpha convolution operator
alpha_edge = EdgeTemplate(name="alpha_edge", operators=[alpha_op])

# create circuit with alpha edge
alpha_net = CircuitTemplate(name="alpha_net", nodes={"li1": li, "li2": li},
                            edges=[("li1/li_op/r", "li2/li_op/m_in", alpha_edge, {"weight": 1.0})])

Since the alpha kernel convolution does not have the same ceiling effect as the hyperbolic tangent, we expect that \(r_2\) increases continuously when we use the same simulation setup as previously. However, the increase should happen more slowly than before, due to the slow time constant of \(\tau_{\alpha} = 10\) that we defined.

res = alpha_net.run(T, dt, inputs={"li1/li_op/u": inp}, outputs={"r1": "li1/li_op/r", "r2": "li2/li_op/r"},
                    solver="scipy", method="RK23")


Indeed, we see that the increase in \(r_2\) is considerably delayed in comparison to the response of \(r_1\). For more options to implement delay coupling in PyRates, see the example on delay coupling in the model definition section.

1.4.3. Example 3: Two-operator edge

In both examples above, we used a rather simple edge template with just a single operator template. However, just as with node templates, it is possible to combine multiple operator templates into a single edge template. To demonstrate this, we will simply combine add a hyperbolic tangent operator to the output of the alpha kernel convolution operator.

combined_edge = EdgeTemplate(name="comb_edge", operators=[alpha_op, tanh_op])

We can just the previously defined operators, since we specified \(x\) as the output variable of alpha_op and as the input variable of tanh_op. Thus, PyRates will know that it should feed the output of the former to the input of the latter.

combined_net = CircuitTemplate(name="alpha_net", nodes={"li1": li, "li2": li},
                               edges=[("li1/li_op/r", "li2/li_op/m_in", combined_edge, {"weight": 5.0})])

The circuit with the combined edge should now yield both effects: A ceiling effect due to the hyperbolic tangent as well as a delayed response due to the alpha kernel.

res = combined_net.run(T, dt, inputs={"li1/li_op/u": inp}, outputs={"r1": "li1/li_op/r", "r2": "li2/li_op/r"},
                       solver="scipy", method="RK23")


…which is what we see as a result!

1.4.4. Example 4: Edges with multiple inputs

As a final example, we will demonstrate how to define edges that receive inputs from multiple sources. A common example for this is the Kuramoto oscillator, where the sinusoidal coupling function depends on difference between the phases of the coupled oscillators. Similarly, we will adapt the edge from example 2 to depend on the difference \(r_1 - r2\) rather than just \(r_1\):

# set up difference-dependent operator
diff_op = OperatorTemplate(name="diff_op",
                           equations=["x' = z/tau",
                                      "z' = r_s - r_t - (2*z + x)/tau"],
                           variables={"x": "output", "z": "variable", "tau": 10.0, "r_s": "input", "r_t": "input"})

# set up edge with the difference-dependent operator
diff_edge = EdgeTemplate(name="diff_edge", operators=[diff_op])

# create circuit where multiple inputs have to be mapped to the edge template
diff_net = CircuitTemplate(name="alpha_net", nodes={"li1": li, "li2": li},
                           edges=[("li1/li_op/r", "li2/li_op/m_in", diff_edge,
                                   {"weight": 1.0,
                                    "diff_edge/diff_op/r_s": "source",
                                    "diff_edge/diff_op/r_t": "li2/li_op/r"})]

As demonstrated above, multiple inputs to an edge template can simply be resolved by specifying which source variable to use for each input in the edge attributes dictionary. Note that source can be used as a keyword to refer to the source variable of the edge (in this case li1/li_op/r). Since this edge only propagates the difference between \(r_1\) and \(r_2\) and only \(r_1\) receives extrinsic input, we expect that \(r_2 < r_1 \forall t\).

res = diff_net.run(T, dt, inputs={"li1/li_op/u": inp}, outputs={"r1": "li1/li_op/r", "r2": "li2/li_op/r"},
                   solver="scipy", method="RK23")


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