|
This section starts by discussing the dynamics of a single neuron and how they change as the input current to the neuron is varied. A network of two coupled neurons is then studied with a particular focus on when the voltage traces of the neurons synchronise. An interesting case is also presented in which the neurons unexpectedly synchronise when there is only a very weak connection between them. Finally the trend observed as the coupling strength is increased is reported. |
Dynamics of a Single Neuron
In this project the Hindmarsh-Rose model is used to model the network. It consists of three coupled differential equations for each neuron and incorporates an external input current. It is of interest to see how the dynamics of the neuron changes as this external input current is varied, and this is firstly done for a single neuron.
Voltage Trace for 2 Amps
|
With an input current of 2.0 amps the voltage trace is regaular with bursts occuring at set intervals, each containing 2 spikes. As the current is increased to 3.2 Amps the trace becomes chaotic and there is no longer a constant interval between spikes. Also there is not a regular number of spikes within each burst. Increasing the input current further causes the trace to again become perioic; now there is always 1 spike within the burst.
In order to demostrate how the trace changes with the input current, the interspike interval can be taken for the trace as a whole and plotted against the input current as the input current is varied. This provides a bifurcation diagram which describes the difference in behaviour of the neuron at different values of input current.
This bifurcation diagram agrees with the traces observed. At 2.0 amps there is a high interspike interval (that between bursts) and a low one (that between the spikes within a group). At 3.2 Amps the diagram indicates the dynamics are chaotic, subsequently stable with 1 spike at 4.0 Amps.
Synchronisation of Two Coupled Neurons
In order to couple two neurons together a coupling strength parameter is introduced. This controls the degree to which one neuron affects the other. Once the neurons are coupled together they gradually synchronise. This means that their voltage traces eventually become the same despite starting at different initial conditions.
By taking the overall time difference between the traces of the neurons as the coupling strength is vaired it is possible to see the point at which the neurons become completely synchronised.
A Blowout Bifurcation
An interesting change in behaviour is observed in Erichsen et. al. as the coupling strength is increased. For a very low coupling strength (0.013) the neurons become completely coherent. As the coupling strength is increased this coherence is lost.
The plots above show this change from complete synchronisation to large deviations away from this as the coupling strength is increased sligtly, from 0.013 to 0.015. This is behaviour typical of a blowout bifurcation.
The Overall Trend
The overall trend is an increase in synchronisation as the coupling strength is increased. The diagram shows burst synchronisation (blue) and spike synchronisation (green). There is a period during which the synchronisation becomes very low (0.15 - 0.2) and this is due to the appearence of an anti-synchronised attrator in this period. The system burst synchronises at around 0.4 and spike synchronises at around 0.47.
Top of Page | Next: More Complex Networks