Dynamics of a Three Neurons Network with Excitatory-Inhibitory Interactions - Foundations on Natural and Artificial Computation - page 98

4.3 Case 3
In figure 6 we present a simulation that gets to a stable point. The parameters
chosen for this simulation are such that we obtain no oscillations. This means
that after some steps the simulation will get to a fixed point on the excitatory-
inhibitory loop where the neurons will be firing or resting constantly. The output
neuron will be always firing or always resting depending on the value of the
threshold.
5 Conclusions
An analytic study of a network consisting of two coupled neurons and an output
neuron is presented here. The model used for the neurons is an Euler approxima-
tion to the leaky integrate and fire. This model is simple enough to do a complete
study of the system dynamic behaviour but suciently detailed to present some
neuronal features as, for example, the effects of inhibition or the generation of
complex rhythm patterns.
This model displays reasonable biological features which may serve, at least,
as a toy model for some neural circuits. We concentrated on extracting the
functional actions rather than reproducing the biological detail.
Acknowledgments
(CA) is supported by MEC BFU2009-08473. (CA) is partially supported by
AYA2009-14212-05. CRL is supported by IBM S.A.E.
References
1. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current
and application to conduction and excitation in nerve. Journal of Physiology 117,
165-181 (1954)
2. Rose, R.M., Hindmarsh, J.L.: The assembly of ionic currents in a thalamic neuron,
I The three dimensional model. Proceedings of The Royal Society of London B 237,
267-288 (1989)
3. Rulkov, N.F.: Modeling of spiking-bursting neural behavior using two-dimensional
map. Physical Review E 65, 041922 (2002)
4. Izhikevich, E.M.: Simple Model of Spiking Neurons. IEEE Trans. on Neural Net-
works 68, 052901 (2003)
5. Shilnikov, A.L., Rulkov, N.F.: Subthreshold oscillations in a map-based neuron
model. Physics Letters A 327, 177-184 (2004)
6. Aguirre, C., Campos, D., Pascual, P., Serrano, E.: Neuronal Behavior with Sub-
threshold Oscillations and Spiking/Bursting Activity Using a Piecewise Linear
Two-Dimensional Map. In: Duch, W., Kacprzyk, J., Oja, E., Zadrozny, S. (eds.)
ICANN 2005. LNCS, vol. 3696, pp. 103-108. Springer, Heidelberg (2005)
7. Kuramoto, Y.: International Symposium on Mathematical Problems in Theoretical
Physics. In: Araki, H. (ed.) Lecture Notes in Physics, vol. 39 (1975)