APS/123-QED Delay-induced destabilization of entrainment of nerve impulses on ephaptically coupled nerve fibers Mohit H. Adhikari and John K. McIver Department of Physics and Astronomy University of New Mexico Evangelos A. Coutsias Department of Mathematics and Statistics University of New Mexico (Dated: October 23, 2008) We study the effect of delay on the synchronization of two nerve impulses traveling along two ephaptically coupled, unmyelinated nerve fibers. The system is modeled as a pair of delay-coupled Fitzhugh-Nagumo equations. A multiple-scale perturbation approach is used for the analysis of these equations in the limit of weak coupling. In the absence of delay, two pulses with identical speeds are shown to be entrained precisely. However, as the delay is increased beyond a critical value, we show that this precise entrainment becomes unstable. We make quantitative estimates for the actual values of delay at which this can occur in the case of squid giant axons and compare them with the relevant time-scales involved. PACS numbers: 02.30.Ks, 87.19.lm, 87.19.lb I. INTRODUCTION Ephaptic coupling refers to interactions between nerve fibers mediated by current flow through the extracellular space without any specialized connecting regions such as synapses for chemical transmission or electronic gap junctions. These interactions occur due to physical proximity of axons, especially those lacking an insulating myelin sheath around them. The mammalian olfactory nerve in which unmyelinated axons are arranged in densely packed fascicles (see figure 1 in [1]) is an exam- ple of a brain region that may favor ephaptic interactions. Experimentally, ephaptic coupling can be detected by observing several phenomena. One such phenomenon is when an action potential on one nerve fiber changes the excitability of the neighboring fibers, and in some cases, evokes action potentials on them. In the second case, adjacent nerve fibers can synchronize their firing pat- terns i.e. the action potentials traveling along them can travel at the same speeds and get phase-locked. These observations were recorded by Katz and Schmitt in 1940 [2] in the case of crab motoneurons by placing two axons in a medium with reduced extracellular conductance. Similar experiments to detect ephaptic coupling have been done using squid giant axons by Arvanitaki [3] and Ramon and Moore [4], active single nerve fibers in the spinal nerve roots of dystrophic mice by Rasminsky [5] and algal strands by Tabata [6]. Early theoretical studies of ephaptic coupling between unmyelinated nerve fibers were done by Markin [7, 8], Luzader and Scott [9], Barr and Plonsey [10]. More recently, Bokil et al. [1] tested the hypothesis that ephaptic interactions occur in a mammalian olfactory nerve by considering the Hodgkin-Huxley model of impulse propagation and showed that an action potential in a single axon can evoke an action potential in all other axons in the fascicle and that the action potentials in neighboring nerve fibers can synchronize. Ephaptic coupling between myelinated nerve fibers whose membrane is covered by a fatty insulating myelin sheath except for some regions called the active nodes, has been studied by Binczak et al. [11] and Reutskiy et al. [12] and Bateman and Van Vleck [13]. Since these interactions occur via the spread of ionic currents, it is reasonable to expect transmission time de- lays in these processes, due to finite times of propagation for these currents. Most experimental papers [3–5] on ephaptic transmission mention that an action potential on one fiber evokes an action potential on an adjacent fiber after a certain time-delay often referred to as the ephaptic transmission time. Ramon and Moore [4] found that this time varied from 200 to 400 µs in the case of squid giant axons. Rasmisky [5] measured an ephaptic transmission time of 100-240 µs in the case of single nerve fibers in the spinal nerve roots of dystrophic mice. While, to our knowledge, there are no experimental measurements of the speeds of the ion currents involved in ephaptic transmission, the observations by Ramon & Moore and Rasminsky strongly suggest the presence of time-delays in ephaptic interactions. Since delays are ubiquitous in dynamical systems and may have profound effects related to stability and the onset of complex behavior [14–17], understanding the impact of ephaptic transmission time-delays on the stability of the entrained state of nerve impulses could be of wider interest. In this article, we study the effect of delay on the entrainment of pulses on two ephaptically coupled, unmyelinated nerve fibers. Our approach is an exten- sion of some old work of Luzader and Scott [9] who developed an analytical model of the ephaptic coupling