### Kuramoto model on a network with realistic cells

Posted:

**Fri Jun 02, 2017 6:22 am**Dear all,

my goal is to construct a network of biophysical (simple) cells and connect them using the kuramoto equation in order to observe phase synchronization of neural oscillations :

https://en.wikipedia.org/wiki/Kuramoto_model

dtheta_{i}/dt=omega _{i}+(K/N)*sum _{j=1...N}*sin(theta _{j}-theta _{i})

N = number of nodes in the network

omega _{i} = natural frequency

theta _{i} = phase

K = coupling strength

Each node in the network is a simple E-I oscillator (or a single cell with one single compartment, this i have to decide). I will keep the natural frequency equal zero or homogeneous over the network and I will keep the coupling strength between each pair of nodes constant.

Right now I am trying to figure out how to implement the Kuramoto model in Neuron. For the connection between cells, I want to write a mod file for a "kuramoto" synapse. The weight of the synapses would represent the coupling strength in the kuramoto model. This mechanism should pass the phase between nodes. As a first step, I am trying to write a function that transforms recorded voltage time series into phases. Analytically this can be done via the Hilbert transform. Large parts of this transformation are already available in hoc via the fft vector method but I am not sure how exactly to implement.

1) Do you think it is suitable to use a synaptic mechanism for implement the Kuramoto model?

I have already constructed a (scale-free) network of biophysical cells. Right now they are connected via excitatory synapses (Exp2syn) which I am going to replace with Kuramoto synapses.

2) Is there already an implementation of the Hilbert transformation in hoc or nmodl?

3) If not should I implement that function as a hoc function, as a nmodl function or as a vector method? I want to be able to call the function from hoc and maybe also from nmodl.

I would appreciate any hints, comments and suggestions that can help me figure out how to construct this model.

my goal is to construct a network of biophysical (simple) cells and connect them using the kuramoto equation in order to observe phase synchronization of neural oscillations :

https://en.wikipedia.org/wiki/Kuramoto_model

dtheta_{i}/dt=omega _{i}+(K/N)*sum _{j=1...N}*sin(theta _{j}-theta _{i})

N = number of nodes in the network

omega _{i} = natural frequency

theta _{i} = phase

K = coupling strength

Each node in the network is a simple E-I oscillator (or a single cell with one single compartment, this i have to decide). I will keep the natural frequency equal zero or homogeneous over the network and I will keep the coupling strength between each pair of nodes constant.

Right now I am trying to figure out how to implement the Kuramoto model in Neuron. For the connection between cells, I want to write a mod file for a "kuramoto" synapse. The weight of the synapses would represent the coupling strength in the kuramoto model. This mechanism should pass the phase between nodes. As a first step, I am trying to write a function that transforms recorded voltage time series into phases. Analytically this can be done via the Hilbert transform. Large parts of this transformation are already available in hoc via the fft vector method but I am not sure how exactly to implement.

1) Do you think it is suitable to use a synaptic mechanism for implement the Kuramoto model?

I have already constructed a (scale-free) network of biophysical cells. Right now they are connected via excitatory synapses (Exp2syn) which I am going to replace with Kuramoto synapses.

2) Is there already an implementation of the Hilbert transformation in hoc or nmodl?

3) If not should I implement that function as a hoc function, as a nmodl function or as a vector method? I want to be able to call the function from hoc and maybe also from nmodl.

I would appreciate any hints, comments and suggestions that can help me figure out how to construct this model.