Stability of synchronous oscillations in a system of Hodgkin-Huxley neurons with delayed diffusive and pulsed coupling Enrico Rossoni * Department of Informatics, Sussex University, Brighton BN1 9QH, United Kingdom Yonghong Chen and Mingzhou Ding Department of Biomedical Engineering, University of Florida, Gainesville, Florida 32611, USA Jianfeng Feng ² Department of Mathematics, Hunan Normal University, 410081 Changsha, People’s Republic of China and Department of Computer Science, Warwick University, Coventry CV4 7AL, United Kingdom Received 3 January 2005; published 9 June 2005 We study the synchronization dynamics for a system of two Hodgkin-Huxley HH neurons coupled diffu- sively or through pulselike interactions. By calculating the maximum transverse Lyapunov exponent, we found that, with diffusive coupling, there are three regions in the parameter space, corresponding to qualitatively distinct behaviors of the coupled dynamics. In particular, the two neurons can synchronize in two regions and desynchronize in the third. When excitatory and inhibitory pulse coupling is considered, we found that syn- chronized dynamics becomes more difficult to achieve in the sense that the parameter regions where the synchronous state is stable are smaller. Numerical simulations of the coupled system are presented to validate these results. The stability of a network of coupled HH neurons is then analyzed and the stability regions in the parameter space are exactly obtained. DOI: 10.1103/PhysRevE.71.061904 PACS numbers: 87.19.La, 05.45.Xt, 02.30.Ks, 87.10.+e I. INTRODUCTION Synchronous oscillations of neuronal activity have been observed at all levels of the nervous system, from the brain- stem to the cortex. The ubiquitous nature of neural oscilla- tions has led to the belief that they may play a key role in information processing. For example, synchronized gamma oscillations have been related to object representation 1, and synchronized neural activity in the somatosensory cortex has been proposed as a mechanism for attentional selection 2. From a theoretical point of view, the problem of under- standing how synchronized oscillations arise has been con- sidered for a variety of systems see 3 for a review. For neuronal systems, theoretical results are usually obtained un- der several simplifying assumptions including instantaneous interactions. However, time delays are inherent in neuronal transmissions because of both finite propagation velocities in the conduction of signals along neurites and delays in the synaptic transmission at chemical synapses 4. It is thus important to understand how synchronization can be achieved when such temporal delays are not negligible 5,6. Indeed, it has been suggested that time delays can actually facilitate synchronization between distant cortical areas. The study of network models has shown that delayed in- teractions can lead to interesting and unexpected phenomena 7. For example, in 8 the authors showed that time delays can induce synchronized periodic oscillations in a network of diffusively coupled oscillators which exhibited chaotic be- havior in the absence of coupling. This was revealed by a stability analysis performed around the synchronized state of the system. Our goal is to examine whether similar results can be found in biophysical neuronal models, such as the Hodgkin- Huxley HH model. Indeed, numerical experiments reported by many authors show that when two systems of the HH type are coupled, they seem to synchronize. Moreover, it has been demonstrated 9 that, in the absence of delay, synchroniza- tion takes place for arbitrary initial conditions for a large class of equations including HH models. For delayed inter- actions, however, an analytical approach to global stability is out of reach, and only local results can be obtained. Here we apply the approach used in 8 to study the stability of the synchronous solutions of coupled HH equations as a function of the coupling strength and time delay. Although the results we obtained are only local, they are still helpful and infor- mative with regards to understanding the mechanisms of synchronization. Moreover, they can be used to reveal re- gions of the parameter space where two neurons cannot syn- chronize, regardless of their initial respective conditions. For two HH neurons coupled diffusively, we found two distinct regions in the parameter space where the synchro- nized dynamics is stable, and one region where it is not stable. These results, based upon the calculation of the maxi- mum transverse Lyapunov exponent, were then confirmed by numerical simulations. The results above are found for neurons with diffusive couplings. Pulse coupled neurons, on the other hand, occur far more frequently in the nervous system 10. Here an ap- proach to tackle the problem of pulse coupling is developed, *Electronic address: e.rossoni@sussex.ac.uk ² Electronic address: jianfeng.feng@warwick.ac.uk PHYSICAL REVIEW E 71, 061904 2005 1539-3755/2005/716/06190411/$23.00 ©2005 The American Physical Society 061904-1