Abstract
Stochastic differential equation (SDE) models of nerve cells for the most part neglect the spatial dimension. Including the latter leads to stochastic partial differential equations (SPDEs) which allow for the inclusion of important variations in the densities of ion channels. In the first part of this work, we briefly consider representations of neuronal anatomy in the context of linear SPDE models on line segments with one and two components. Such models are reviewed and analytical methods illustrated for finding solutions as series of Ornstein–Uhlenbeck processes. However, only nonlinear models exhibit natural spike thresholds and admit traveling wave solutions, so the rest of the article is concerned with spatial versions of the two most studied nonlinear models, the Hodgkin–Huxley system and the FitzHugh–Nagumo approximation. The ion currents underlying neuronal spiking are first discussed and a general nonlinear SPDE model is presented. Guided by recent results for noise-induced inhibition of spiking in the corresponding system of ordinary differential equations, in the spatial Hodgkin–Huxley model, excitation is applied over a small region and the spiking activity observed as a function of mean stimulus strength with a view to finding the critical values for repetitive firing. During spiking near those critical values, noise of increasing amplitudes is applied over the whole neuron and over restricted regions. Minima have been found in the spike counts which parallel results for the point model and which have been termed inverse stochastic resonance. A stochastic FitzHugh–Nagumo system is also described and results given for the probability of transmission along a neuron in the presence of noise.
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References
Austin, T.D.: The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism. Ann. Appl. Probab. 18, 1279–1325 (2008)
Bergé, B., Chueshov, I.D., Vuillermot, P.A.: On the behavior of solutions to certain parabolic SPDEs driven by Wiener processes. Stoch. Proc. Appl. 92, 237–263 (2001)
Burlhis, T.M., Aghajanian, G.K.: Pacemaker potentials of serotonergic dorsal raphe neurons: contribution of a low-threshold Ca2 + conductance. Synapse 1, 582–588 (1987)
Destexhe, A., Sejnowski, O.: Thalamocortical Assemblies. Oxford University Press, Oxford (2001)
Ditlevsen, S., Ditlevsen, O.: Parameter estimation from observations of first-passage times of the Ornstein-Uhlenbeck process and the Feller process. Probabilist. Eng. Mech. 23, 170–179 (2008)
Ditlevsen, S., Lansky, P.: Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model. Phys. Rev. E 71, Art. No. 011,907 (2005)
Dodge, F.A., Cooley, J.: Action potential of the motoneuron. IBM J. Res. Devel. 17, 219–229 (1973)
Dolphin, A.C.: Calcium channel diversity: multiple roles of calcium channel subunits. Curr. Opin. Neurobiol. 19, 237–244 (2009)
FitzHugh, R.: Mathematical models of excitation and propagation in nerve. In: Biological Engineering. McGrawHill, New York (1969)
Gerstein, G.L., Mandelbrot, B.: Random walk models for the spike activity of a single neuron. Biophys. J. 4, 4168 (1964)
Gluss, B.: A model for neuron firing with exponential decay of potential resulting in diffusion equations for probability density. Bull. Math. Biophys. 29, 233–243 (1967)
Goldfinger, M.D.: Poisson process stimulation of an excitable membrane cable model. Biophys. J. 50, 27–40 (1986)
Gutkin, B.S., Jost, J., Tuckwell, H.C.: Inhibition of rhythmic neural spiking by noise: the occurrence of a minimum in activity with increasing noise. Naturwissenschaften 96, 1091–1097 (2009)
Gutman, G.A., Chandy, K.G., Grissmer, S., Lazdunski, M., McKinnon, D., Pardo, L.A., Robertson, G.A., Rudy, B., Sanguinetti, M.C., Stuhmer, W., Wang, X.: International Union of Pharmacology. LIII. Nomenclature and molecular relationships of voltage-gated potassium channels. Pharmacol. Rev. 57, 473,508 (2005)
Hanson, F.B., Tuckwell, H.C.: Diffusion approximations for neuronal activity including synaptic reversal potentials. J. Theoret. Neurobiol. 2, 127–153 (1983)
Hellwig, B.: A quantitative analysis of the local connectivity between pyramidal neurons in layers 2/3 of the rat visual cortex. Biol. Cybern. 82, 111–121 (2000)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)
Horikawa, Y.: Noise effects on spike propagation in the stochastic Hodgkin-Huxley models. Biol. Cybern. 66, 19–25 (1991)
Iannella, N., Tanaka, S., Tuckwell, H.C.: Firing properties of a stochastic PDE model of a rat sensory cortex layer 2/3 pyramidal cell. Math. Biosci. 188, 117–132 (2004)
Kallianpur, G., Xiong, J.: Diffusion approximation of nuclear space-valued stochastic differential equations driven by Poisson random measures. Ann. Appl. Probab. 5, 493–517 (1995)
Koch, C.: Biophysics of Computation: Information Processing in Single Neurons. Oxford University Press, Oxford (1999)
Komendantov, A.O., Tasker, J.G., Trayanova, N.A.: Somato-dendritic mechanisms underlying the electrophysiological properties of hypothalamic magnocellular neuroendocrine cells: A multicompartmental model study. J. Comput. Neurosci. 23, 143–168 (2007)
Levitan, I.B., Kaczmarek, L.K.: Neuromodulation. Oxford University Press, Oxford (1987)
Lindner, B., Garcia-Ojalvo, J., Neiman, A., Schimansky-Geier, L.: Effects of noise in excitable systems. Phys. Rep. 392, 321–424 (2004)
Llinas, R.: The intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function. Science 242, 1654–1664 (1988)
Mainen, Z.F., Joerges, J., Huguenard, J.R., Sejnowski, T.J.: A model of spike initiation in neocortical pyramidal neurons. Neuron 15, 1427–1439 (1995)
Markram, H., Toledo-Rodriguez, M., Wang, Y., Gupta, A., Silberberg, G., Wu, C.: Interneurons of the neocortical inhibitory system. Nat. Rev. Neurosci. 5, 793–807 (2004)
McCormick, D.A., Huguenard, J.R.: A model of the electrophysiological properties of thalamocortical relay neurons. J. Neurophysiol. 68, 1384–1400 (1992)
Megías, M., Emri, Z.S., Freund, T.F., Gulyás, A.I.: Total number and distribution of inhibitory and excitatory synapses on hippocampal CA1 pyramidal cells. Neuroscience 102, 527–540 (2001)
Meir, A., Ginsburg, S., Butkevich, A., Kachalsky, S.G., Kaiserman, I., Ahdut, R., Demirgoren, S., Rahamimoff, R.: Ion channels in presynaptic nerve terminals and control of transmitter release. Physiol. Rev. 79, 1020–1088 (1999)
Rhodes, P.A., Llinas, R.: A model of thalamocortical relay cells. J. Physiol. 565, 765–781 (2005)
Roy, B.K., Smith, D.R.: Analysis of the exponential decay model of the neuron showing frequency threshold effects. Bull. Math. Biophys. 31, 341–357 (1969)
Sholl, D.: The Organization of the Cerebral Cortex. Methuen, London (1956)
Shu, Y., Hasenstaub, A., Badoual, M., Bal, T., McCormick, D.A.: Barrages of synaptic activity control the gain and sensitivity of cortical neurons. J. Neurosci. 23, 10388–10401 (2003)
Skaugen, E., Walloe, L.: Firing behaviour in a stochastic nerve membrane model based upon the Hodgkin-Huxley equations. Acta Physiol. Scand. 107, 343–363 (1979)
Spruston, N.: Pyramidal neurons: dendritic structure and synaptic integration. Nat. Rev. Neurosci. 9, 206–221 (2008)
Traub, R.D.: Motoneurons of different geometry and the size principle. Biol. Cybern. 25, 163–175 (1977)
Traub, R.D.: Neocortical pyramidal cells: a model with dendritic calcium conductance reproduces repetitive firing and epileptic behavior. Brain Res. 173, 243–257 (1979)
Tuckwell, H.C.: Synaptic transmission in a model for stochastic neural activity. J. Theor. Biol. 77, 65–81 (1979)
Tuckwell, H.C.: Poisson Processes in Biology. In: Stochastic Nonlinear Systems, pp. 162–172. Springer, Berlin (1981)
Tuckwell, H.C.: Stochastic equations for nerve membrane potential. J. Theoret. Neurobiol. 5, 87–99 (1986)
Tuckwell, H.C.: Introduction to Theoretical Neurobiology, vol. 1: Linear Cable Theory and Dendritic Structure. Cambridge University Press, Cambridge (1988)
Tuckwell, H.C.: Introduction to Theoretical Neurobiology, vol. 2: Nonlinear and Stochastic Theories. Cambridge University Press, Cambridge (1988)
Tuckwell, H.C.: Stochastic Processes in the Neurosciences. SIAM, Philadelphia (1989)
Tuckwell, H.C.: Spatial neuron model with two-parameter Ornstein-Uhlenbeck input current. Phys. A 368, 495–510 (2006)
Tuckwell, H.C.: Analytical and simulation results for the stochastic spatial FitzHugh-Nagumo neuron. Neural Comput. 20, 3003–3035 (2008)
Tuckwell, H.C., Jost, J.: Weak noise in neurons may powerfully inhibit the generation of repetitive spiking but not its propagation. PLoS Comp. Biol. 6, e1000794 (2010)
Tuckwell, H.C., Jost, J.: The effects of various spatial distributions of weak noise on rhythmic spiking. J. Comp. Neurosci. 30, 361–371 (2011)
Tuckwell, H.C., Walsh, J.B.: Random currents through nerve membranes. Biol. Cybern. 49, 99–110 (1983)
Tuckwell, H.C., Wan, F.Y.M., Wong, Y.S.: The interspike interval of a cable model neuron with white noise input. Biol. Cybern. 49, 155–167 (1984)
Tuckwell, H.C., Wan, F.Y.M., Rospars, J.P.: A spatial stochastic neuronal model with Ornstein-Uhlenbeck input current. Biol. Cybern. 86, 137–145 (2002)
Tuckwell, H.C., Jost, J., Gutkin, B.S.: Inhibition and modulation of rhythmic neuronal spiking by noise. Phys. Rev. E 80, 031907 (2009)
Watts, J., Thomson, A.M.: Excitatory and inhibitory connections show selectivity in the neocortex. J. Physiol. 562.1, 89–97 (2005)
Zhang, X., You, G., Chen, T., Feng, J.: Maximum likelihood decoding of neuronal inputs from an interspike interval distribution. Neural Comput. 21, 1–27 (2009)
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Tuckwell, H.C. (2013). Stochastic Partial Differential Equations in Neurobiology: Linear and Nonlinear Models for Spiking Neurons. In: Bachar, M., Batzel, J., Ditlevsen, S. (eds) Stochastic Biomathematical Models. Lecture Notes in Mathematics(), vol 2058. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32157-3_6
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