Analysis of Synchronization in Pulse Neural Networks with Phase Response Function
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We treat two oscillatin neurons interacting with weak couplings.
For simplicity, let us define the dynamics of one neuron as
Explanation of Applet
The dependence of the stationary phase difference Δφ (denoted as "p") on the synaptic rate α is shown.
The red lines show the stable phase differences,
and the blue lines show the unstable phase differences.
The form of lines changes according to the neuron models.
By clicking the field, you can change the value of synaptic rate α
(The pink line show the present value of α.).
By clicking the outside of the field, the pink line disappears.
V(φ), Z(φ), I(φ)
The membrane potential V(φ) of the neuron,
the phase response Z(φ),
and the synaptic input I(φ) are shown.
The waveform of I(φ) changes according to the value of
the synaptic rate α.
You can regard the phase φ as the time t.
Γ(φ) and Γodd(φ) calculated from Z(φ) and I(φ) are shown.
The intersections of Γodd(φ) and the horizontal axis
show the stationary phase differences Δφ,
and the red circles and
the blue circles show the stable and unstable
phase differences, respectively.
This graph corresponds to the graph in (α,Δφ) plane.
Simulation of the network with 100 neurons.
When the synchronized solution (Δφ=0) is stable,
all neurons tend to synchronize with each other.
where f(x) governs the dynamic of neuron x,
and I(x,x') is the synaptic input from the other neuron.
The synaptic inputs are often modeled by a sum of exponential functions
or alpha functions.
where tk is the k-th firing time of this neuron
and α is the synaptic rate.
Generally, weakly coupled two oscillators can be reduced to the
phase model written as
where Z(φ) is the phase response function
that shows the change in frequency caused by a small perturbation.
After averaging, the dynamics of weakly coupled two neurons
follow the phase model written as
where Γ(φ) can be calculated by the following formula.
The dynamics of the phase difference Δφ of two oscillating neurons
can be analyzed by investigating the following model.
Note that the stationary phase difference Δφ
and its stability can be determined by the sign of the slope.
You can perform analyses of synchronization in pulse neural networks
using the phase response function.
You can choose various neuron models and synaptic inputs.
Pulse neural networks are composed of 100 neurons, and their
connections are global as follows.
- Models: Leaky Integrator / Hodgkin-Huxley (class2) / FitzHugh-Nagumo (class 2) / Morris-Lecar (class1)
- Synaptic inputs: Exponential / Alpha (You can also change the sign of inputs)
When the stationary phase difference with Δφ=0 is stable,
neurons tend to fire synchronously.
The definitions of neuron models are shown
in this page.
This page is based on the following papers.
- Takashi Kanamaru
"Analysis of synchronization between two modules of pulse neural networks with excitatory and inhibitory connections,"
Neural Computation, vol.18, no.5 (2006) pp.1111-1131.
- Bard Ermentrout
"Type I membranes, phase resetting curves, and synchrony,"
Neural Computation, 18, 979-1001 (1996)
- D. Hansel, G. Mato, and C. Meunier
"Synchrony in excitatory neural networks,"
Neural Computation, 7, 307-337 (1995)