## Waves and bumps in neuronal networks with axo-dendritic synaptic interactions.(English)Zbl 1013.92006

Summary: We consider a firing rate model of a neuronal network continuum that incorporates axo-dendritic synaptic processing and the finite conduction velocities of action potentials. The model equation is an integral one defined on a spatially extended domain. Apart from a spatial integral mixing, the network connectivity function with space-dependent delays, arising from non-instantaneous axonal communication, the integral model also includes a temporal integration over some appropriately identified distributed delay kernels. These distributed delay kernels are biologically motivated and represent the response of biological synapses to spiking inputs. They are interpreted as Green’s functions of some linear differential operators.
Exploiting this Green’s function description, we discuss formal reductions of this non-local system to equivalent partial differential equation (PDE) models. We distinguish between those spatial connectivity functions that give rise to local PDE models and those that give rise to PDE models with delayed non-local terms. For cases in which local PDEs are derived, we investigate traveling wave solutions in a co-moving frame by numerically computing global heteroclinic connections for sigmoidal firing rate functions.
We also calculate exact solutions, parameterized by axonal conduction velocity, for the Heaviside firing rate function (the sigmoidal firing rate function in the limit of infinite gain). The inclusion of synaptic adaptation is shown to alter traveling wave fronts to traveling pulses, which we study analytically and numerically in terms of a global homoclinic orbit. Finally, we consider the impact of dendritic interactions on waves and on static spatially localized solutions. Exact analysis for infinite gain shows that axonal delays do not affect the stability of single bumps. Furthermore, numerical continuation for finite gain leads to multiple bump solutions, and it is demonstrated that such localized multi-bumps are lost (in favor of global patterns) when a stable $$N$$-bump and an unstable $$(N+2)$$-bump coalesce.
In essence, the work in this paper illustrates how physiologically significant features of synaptic processing, synaptic adaptation and patterns of axo-dendritic connectivity may be analyzed within a neural field firing rate framework.

### MSC:

 92C20 Neural biology 35Q92 PDEs in connection with biology, chemistry and other natural sciences

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 [1] Golomb, D.; Amitai, Y., Propagating neuronal discharges in neocortical slices: computational and experimental study, J. neurophysiol., 78, 1199-1211, (1997) [2] Miles, R.; Traub, R.D.; Wong, R.K.S., Spread of synchronous firing in longitudinal slices from the CA3 region of hippocampus, J. neurophysiol., 60, 1481-1496, (1995) [3] Kim, U.; Bal, T.; McCormick, D.A., Spindle waves are propagating synchronized oscillations in the ferret lgnd in vitro, J. neurophysiol., 74, 1301-1323, (1995) [4] Roberts, A., How does a nervous system produce behaviour? A case study in neurobiology, Sci. prog. Oxford, 74, 31-51, (1990) [5] B.W. Connors, Y. Amitai, Generation of epileptiform discharges by local circuits in neocortex, in: P.A. Schwartzkroin (Ed.), Epilepsy: Models, Mechanisms and Concepts, Cambridge University Press, Cambridge, 1993, pp. 388-424. [6] Gray, C.M., Synchronous oscillations in neuronal systems: mechanisms and functions, J. comput. neurosci., 1, 11-38, (1992) [7] Colby, C.L.; Duhamel, J.R.; Goldberg, M.E., Occulocentric spatial representation in parietal cortex, Cereb. cortex, 5, 470-481, (1995) [8] Golomb, D.; Wang, X.J.; Rinzel, J., Propagation of spindle waves in a thalamic slice model, J. neurophysiol., 75, 750-769, (1996) [9] Rinzel, J.; Terman, D.; Wang, X.J.; Ermentrout, B., Propagating activity patterns in large-scale inhibitory neuronal networks, Science, 279, 1351-1355, (1998) [10] Pinto, D.J.; Ermentrout, G.B., Spatially structured activity in synaptically coupled neuronal networks. I. travelling fronts and pulses, SIAM J. appl. math., 62, 206-225, (2001) · Zbl 1001.92021 [11] C.R. Laing, W.C. Troy, B. Gutkin, G.B. Ermentrout, Multiple bumps in a neuronal model of working memory, SIAM J. Appl. Math. 63 (2002) 62-97. · Zbl 1017.45006 [12] Pinto, D.J.; Ermentrout, G.B., Spatially structured activity in synaptically coupled neuronal networks. II. lateral inhibition and standing pulses, SIAM J. appl. math., 62, 226-243, (2001) · Zbl 1070.92506 [13] J. Huguenard, D.A. McCormick, Electrophysiology of the neuron, Oxford University Press, Oxford, 1996. [14] Wilson, H.R.; Cowan, J.D., A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13, 55-80, (1973) · Zbl 0281.92003 [15] Nunez, P.L., The brain wave equation: a model for the EEG, Math. biosci., 21, 279, (1974) · Zbl 0292.92001 [16] Amari, S, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. cybern., 27, 77-87, (1977) · Zbl 0367.92005 [17] Jirsa, V.K.; Haken, H., Field theory of electromagnetic brain activity, Phys. rev. lett., 77, 960-963, (1996) [18] Bressloff, P.C.; Coombes, S., Dynamics of strongly-coupled spiking neurons, Neural comput., 12, 91-129, (2000) [19] Ermentrout, G.B., Neural nets as spatio-temporal pattern forming systems, Rep. prog. phys., 61, 353-430, (1998) [20] J. Rinzel, G.B. Ermentrout, Analysis of neural excitability and oscillations, in: C. Koch, I. Segev (Eds.), Methods in Neuronal Modeling, MIT Press, Cambridge, MA, 1989, pp. 135-170. [21] E.L. White, Cortical Circuits: Synaptic Organization of the Cerebral Cortex. Structure, Function, and Theory, Birkhauser, Basel, 1989. [22] Ermentrout, G.B.; McLeod, J.B., Existence and uniqueness of travelling waves for a neural network, Proc. roy. soc. Edinburgh A, 123, 461-478, (1993) · Zbl 0797.35072 [23] K. Schumacher, Travelling-front solutions for integrodifferential equations II, in: W. Jäger, H. Rost, P. Tautu (Eds.), Lecture Notes in Biomathematics: Biological Growth and Spread, vol. 38, Springer, Berlin, 1980. · Zbl 0419.45004 [24] Kelso, J.A.S.; Bressler, S.L.; Buchanan, S.; Deguzman, G.C.; Ding, M.; Fuchs, A.; Holroyd, T., A phase-transition in human brain and behaviour, Phys. lett. A, 169, 134-144, (1992) [25] Wu, J.; Zou, X., Asymptotic and periodic boundary value problems of mixed FDEs and wave solutions of lattice differential equations, J. diff. eqs., 135, 315-357, (1997) · Zbl 0877.34046 [26] Elmer, C.E.; Van Vleck, E.S., Analysis and computation of travelling wave solutions of bistable differential-difference equations, Nonlinearity, 12, 771-798, (1999) · Zbl 0945.35046 [27] E.J. Doedel, A.R. Champneys, T.R. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X.J. Wang, AUTO97 Continuation and bifurcation software for ordinary differential equations, 1997. http://indy.cs.concordia.ca/auto/main.html. [28] Beyn, W.J., The numerical computation of connecting orbits in dynamical systems, IMA J. numer. anal., 9, 379-405, (1990) · Zbl 0706.65080 [29] Doedel, E.; Friedman, M.J.; Kunin, B.I., Successive continuation for locating connecting orbits, Numer. algorithms, 14, 103-124, (1997) · Zbl 0885.65080 [30] Sandstede, B., Convergence estimates for the numerical approximation of homoclinic solutions, IMA J. numer. anal., 17, 437-462, (1997) · Zbl 0899.65044 [31] Liu, Y.H.; Wang, X.J., Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron, J. comput. neurosci., 10, 25-45, (2001) [32] Bressloff, P.C., New mechanism for neural pattern formation, Phys. rev. lett., 76, 4644-4647, (1996) [33] J.D. Murray, Mathematical Biology, Springer, Berlin, 1989. · Zbl 0682.92001 [34] P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer, Berlin, 1979. · Zbl 0403.92004 [35] Bressloff, P.C.; Coombes, S., Physics of the extended neuron, Int. J. mod. phys. B, 11, 2343-2392, (1997) [36] Cross, M.C.; Hohenberg, P.C., Pattern formation outside of equilibrium, Rev. mod. phys., 65, 851-1112, (1993) · Zbl 1371.37001 [37] Champneys, A.R., Homoclinic orbits in reversible systems and their applications in mechanics, Physica D, 112, 158-186, (1998) · Zbl 1194.37154 [38] Woods, P.D.; Champneys, A.R., Heteroclinic tangles in the unfolding of a degenerate Hamiltonian Hopf bifurcation, Physica D, 129, 147-170, (1999) · Zbl 0952.37009 [39] Hunt, G.W.; Lord, G.J.; Champneys, A.R., Homoclinic and heteroclinic orbits underlying the post-buckling of axially compressed cylindrical shells, Comput. meth. appl. mech. eng., spec. issue comput. meth. bifurcat. theor., 170, 239-251, (1999) · Zbl 0958.74021 [40] Hunt, G.W.; Peletier, M.A.; Champneys, A.R.; Woods, P.D.; Ahmer Wadee, M.; Budd, C.J.; Lord, G.J., Cellular buckling in long structures, Theme issue localization, 21, 3-29, (2000) · Zbl 0974.74024 [41] Swindale, N.V., The development of topography in the visual cortex: a review of models, Network, 7, 161-274, (1996) · Zbl 0903.92012 [42] Bressloff, P.C., Traveling fronts and wave propagation failure in an inhomogeneous neural network, Physica D, 155, 83-100, (2001) · Zbl 1004.92005 [43] D. Terman, L. Zhang, Asymptotic stability of traveling wave solutions of integral – differential equations arising from neuronal networks, preprint, 2002. http://www.math.ohio-state.edu/∼terman/. [44] Taylor, J.G., Neural bubble dynamics in two dimensions: foundations, Biol. cybernet., 80, 393-409, (1999) · Zbl 0984.92009 [45] Werner, H.; Richter, T., Circular stationary solutions in two-dimensional neural fields, Biol. cybernet., 85, 211-217, (2001) · Zbl 1160.92323 [46] Li, C.; Jasper, H.H., Microelectrode studies of the electrical activity of the cerebral cortex in the cast, J. physiol., 121, 117-140, (1953) [47] Koch, C., Cable theory in neurons with active, linearized membranes, Biol. cybernet., 50, 15-33, (1984) [48] Segev, I.; Rall, W., Excitable dendrites and spines: earlier theoretical insights elucidate recent direct observations, Trends neurosci., 21, 11, 453-460, (1998) [49] Bressloff, P.C.; Coombes, S., Spike train dynamics underlying pattern formation in integrate-and-fire oscillator networks, Phys. rev. lett., 81, 2384-2387, (1998) [50] Laing, C.; Chow, C.C., Stationary bumps in networks of spiking neurons, Neural comput., 13, 1473-1494, (2001) · Zbl 0978.92004
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