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Some Joys and Trials of Mathematical Neuroscience

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An Erratum to this article was published on 12 March 2014

Abstract

I describe the basic components of the nervous system—neurons and their connections via chemical synapses and electrical gap junctions—and review the model for the action potential produced by a single neuron, proposed by Hodgkin and Huxley (HH) over 60 years ago. I then review simplifications of the HH model and extensions that address bursting behavior typical of motoneurons, and describe some models of neural circuits found in pattern generators for locomotion. Such circuits can be studied and modeled in relative isolation from the central nervous system and brain, but the brain itself (and especially the human cortex) presents a much greater challenge due to the huge numbers of neurons and synapses involved. Nonetheless, simple stochastic accumulator models can reproduce both behavioral and electrophysiological data and offer explanations for human behavior in perceptual decisions. In the second part of the paper I introduce these models and describe their relation to an optimal strategy for identifying a signal obscured by noise, thus providing a norm against which behavior can be assessed and suggesting reasons for suboptimal performance. Accumulators describe average activities in brain areas associated with the stimuli and response modes used in the experiments, and they can be derived, albeit non-rigorously, from simplified HH models of excitatory and inhibitory neural populations. Finally, I note topics excluded due to space constraints and identify some open problems.

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Notes

  1. Biologists refer to dendrites and axons as processes: confusing terminology for a mathematician!

  2. Division by 103 accommodates the conventional units of millivolts, nanoamps, and nanosiemens for conductances.

References

  • Abbott, L.F.: Theoretical neuroscience rising. Neuron 60, 489–495 (2008)

    Google Scholar 

  • Arnold, L.: Stochastic Differential Equations. Wiley, New York (1974)

    MATH  Google Scholar 

  • Aston-Jones, G., Cohen, J.D.: An integrative theory of locus coeruleus-norepinephrine function: adaptive gain and optimal performance. Annu. Rev. Neurosci. 28, 403–450 (2005)

    Google Scholar 

  • Ayali, A., Fuchs, E., Hulata, E., Jacob, E.B.: The function of inter segmental connections in determining temporal characteristics of the spinal cord rhythmic output. Neuroscience 147, 236–246 (2007)

    Google Scholar 

  • Back, A., Guckenheimer, J., Myers, M.: A dynamical simulation facility for hybrid systems. In: Lecture Notes in Computer Science, vol. 736, pp. 255–267. Springer, Berlin (1993)

    Google Scholar 

  • Baker, S.N.: Oscillatory interactions between sensorimotor cortex and periphery. Curr. Opin. Neurobiol. 17, 649–655 (2007)

    Google Scholar 

  • Balan, P.F., Gottlieb, J.: Functional significance of nonspatial information in monkey lateral intraparietal area. J. Neurosci. 29(25), 8166–8176 (2009)

    Google Scholar 

  • Balci, F., Simen, P., Niyogi, R., Saxe, A., Holmes, P., Cohen, J.D.: Acquisition of decision making criteria: reward rate ultimately beats accuracy. Atten. Percept. Psychophys. 73(2), 640–657 (2011)

    Google Scholar 

  • Bayes, T.: An essay toward solving a problem in the doctrine of chances. Philos. Trans. R. Soc. Lond. 53, 370–418 (1763)

    Google Scholar 

  • Bejjanki, V.R., Lu, Z.L., Beck, J.M., Pouget, A.: Perceptual learning as improved probabilistic inference in early sensory areas. Nat. Neurosci. 14(6), 783–790 (2011)

    Google Scholar 

  • Ben-Haim, Y.: Information Gap Decision Theory: Decisions Under Severe Uncertainty, 2nd edn. Academic Press, New York (2006)

    Google Scholar 

  • Berridge, C.W., Waterhouse, B.D.: The locus coeruleus-noradrenergic system: modulation of behavioral state and state-dependent cognitive processes. Brains Res. Rev. 42, 33–84 (2003)

    Google Scholar 

  • Bogacz, R., Gurney, K.: The basal ganglia and cortex implement optimal decision making between alternative actions. Neural Comput. 19, 442–477 (2007)

    MATH  MathSciNet  Google Scholar 

  • Bogacz, R., Larsen, T.: Integration of reinforcement learning and optimal decision-making theories of the basal ganglia. Neural Comput. 23, 817–851 (2011)

    MATH  MathSciNet  Google Scholar 

  • Bogacz, R., Brown, E., Moehlis, J., Holmes, P., Cohen, J.D.: The physics of optimal decision making: a formal analysis of models of performance in two alternative forced choice tasks. Psychol. Rev. 113(4), 700–765 (2006)

    Google Scholar 

  • Bogacz, R., Hu, P., Holmes, P., Cohen, J.D.: Do humans produce the speed-accuracy tradeoff that maximizes reward rate? Q. J. Exp. Psychol. 63(5), 863–891 (2010)

    Google Scholar 

  • Bohil, C.J., Maddox, W.T.: On the generality of optimal versus objective classifier feedback effects on decision criterion learning in perceptual categorization. Mem. Cogn. 31(2), 181–198 (2003)

    Google Scholar 

  • Botvinick, M.M.: Hierarchical reinforcement learning and decision making. Curr. Opin. Neurobiol. 22, 956–962 (2012)

    Google Scholar 

  • Botvinick, M.M., Niv, Y., Barto, A.C.: Hierarchically organized behavior and its neural foundations: a reinforcement-learning perspective. Cognition 113, 262–280 (2009)

    Google Scholar 

  • Britten, K.H., Shadlen, M.N., Newsome, W.T., Movshon, J.A.: Responses of neurons in macaque MT to stochastic motion signals. Vis. Neurosci. 10, 1157–1169 (1993)

    Google Scholar 

  • Brown, E., Moehlis, J., Holmes, P.: On the phase reduction and response dynamics of neural oscillator populations. Neural Comput. 16(4), 673–715 (2004)

    MATH  Google Scholar 

  • Brown, E., Gao, J., Holmes, P., Bogacz, R., Gilzenrat, M., Cohen, J.: Simple networks that optimize decisions. Int. J. Bifurc. Chaos 15(3), 803–826 (2005)

    MATH  MathSciNet  Google Scholar 

  • Brunel, N., van Rossum, M.C.W.: Lapicque’s 1907 paper: from frongs to integrate-and-fire. Biol. Cybern. 97, 337–339 (2007)

    MATH  Google Scholar 

  • Brunel, N., Wang, X.-J.: Effects of neuromodulation in a cortical network model. J. Comput. Neurosci. 11, 63–85 (2001)

    Google Scholar 

  • Buchanan, J.T., Grillner, S.: Newly identified glutamate interneurons and their role in locomotion in the lamprey spinal cord. Science 236, 312–314 (1987)

    Google Scholar 

  • Buhusi, C.V., Meck, W.H.: What makes us tick? Functional and neural mechanisms of interval timing. Nature 6, 755–765 (2005)

    Google Scholar 

  • Burkitt, A.N.: A review of the integrate-and-fire neuron model: I. Homogeneous synaptic input. Biol. Cybern. 95, 1–19 (2006a)

    MATH  MathSciNet  Google Scholar 

  • Burkitt, A.N.: A review of the integrate-and-fire neuron model: II. Inhomogeneous synaptic input and network properties. Biol. Cybern. 95, 97–112 (2006b)

    MATH  MathSciNet  Google Scholar 

  • Busemeyer, J.R., Townsend, J.T.: Decision field theory: a dynamic-cognitive approach to decision making in an uncertain environment. Psychol. Rev. 100, 432–459 (1993)

    Google Scholar 

  • Buzsaki, G.: Rhythms of the Brain. Oxford University Press, New York (2006)

    MATH  Google Scholar 

  • Chay, T.R., Keizer, J.: Minimal model for membrane oscillations in the pancreatic β-cell. Biophys. J. 42, 181–190 (1983)

    Google Scholar 

  • Chiel, H.J., Beer, R.D.: The brain has a body: adaptive behavior emerges from interactions of nervous system, body and environment. Trends Neurosci. 20(12), 553–557 (1997)

    Google Scholar 

  • Cho, R.Y., Nystrom, L.E., Brown, E., Jones, A.D., Braver, T.S., Holmes, P., Cohen, J.D.: Mechanisms underlying dependencies of performance on stimulus history in a two-alternative forced-choice task. Cogn. Affect. Behav. Neurosci. 2(4), 283–299 (2002)

    Google Scholar 

  • Chung, K., Wallace, J., Kim, S.-Y., Kalyanasundaram, S., Andalman, A.S., Davidson, T.J., Mirzabekov, J.J., Zalocusky, K.A., Mattis, J., Denisin, A.K., Pak, S., Bernstein, H., Ramakrishnan, C., Grosenick, L., Gradinaru, V., Deisseroth, K.: Structural and molecular interrogation of intact biological systems. Nature 497, 332–337 (2013)

    Google Scholar 

  • Cohen, A.H., Holmes, P., Rand, R.H.: The nature of coupling between segmental oscillators of the lamprey spinal generator for locomotion: a model. J. Math. Biol. 13, 345–369 (1982)

    MATH  MathSciNet  Google Scholar 

  • Cohen, A.H., Rossignol, S., Grillner, S. (eds.): Neural Control of Rhythmic Movements in Vertebrates. Wiley, New York (1988)

    Google Scholar 

  • Cohen, J.D., Dunbar, K., McClelland, J.L.: On the control of automatic processes: a parallel distributed processing model of the Stroop effect. Psychol. Rev. 97(3), 332–361 (1990)

    Google Scholar 

  • Cohen, J.D., Servan-Schreiber, D., McClelland, J.L.: A parallel distributed processing approach to automaticity. Am. J. Psychol. 105, 239–269 (1992)

    Google Scholar 

  • Coulthard, E.J., Bogacz, R., Javed, S., Mooney, L.K., Murphy, G., Keeley, S., Whone, A.L.: Distinct roles of dopamine and subthalamic nucleus in learning and probabilistic decision making. Brain 135, 3721–3734 (2012)

    Google Scholar 

  • Daun-Gruhn, S.: A mathematical modeling study of inter-segmental coordination during stick insect walking. J. Comput. Neurosci. 30(2), 255–278 (2011)

    MathSciNet  Google Scholar 

  • Daun-Gruhn, S., Toth, T.I.: An inter-segmental network model and its use in elucidating gait-switches in the stick insect. J. Comput. Neurosci. 31(1), 43–60 (2011)

    Google Scholar 

  • Daun-Gruhn, S., Rubin, J.E., Rybak, I.A.: Control of oscillation periods and phase durations in half-center central pattern generators: a comparative mechanistic analysis. J. Comput. Neurosci. 27(1), 3–36 (2009)

    MathSciNet  Google Scholar 

  • Dayan, P.: How to set the switches on this thing. Curr. Opin. Neurobiol. 22, 1068–1074 (2012)

    Google Scholar 

  • Dayan, P., Abbott, L.F.: Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. MIT Press, Cambridge (2001)

    Google Scholar 

  • De Schutter, E.: Why are computational neuroscience and systems biology so separate? PLoS Comput. Biol. 4(5), e1000078 (2008)

    Google Scholar 

  • De Schutter, E. (ed.): Computational Modeling Methods for Neuroscientists. MIT Press, Cambridge (2009)

    Google Scholar 

  • Deco, G., Rolls, E.T., Albantakis, L., Romo, R.: Brain mechanisms for perceptual and reward-related decision-making. Prog. Neurobiol. 103, 194–213 (2013)

    Google Scholar 

  • Destexhe, A., Mainen, Z.F., Sejnowski, T.J.: Kinetic models of synaptic transmission. In: Koch, C., Segev, I. (eds.) Methods in Neuronal Modeling: From Ions to Networks, 2nd edn., pp. 1–25. MIT Press, Cambridge (1999)

    Google Scholar 

  • Doloc-Mihu, A., Calabrese, R.L.: A database of computational models of a half-center oscillator for analyzing how neuronal parameters influence network activity. J. Biol. Phys. 37(3), 263–283 (2011)

    Google Scholar 

  • Doya, K., Shadlen, M.N. (eds.): Decision making. Curr. Opin. Neurobiol. 22(6) (2012). Special issue

  • Dutilh, G., Vandekerckhove, J., Tuerlinckx, F., Wagenmakers, E.J.: A diffusion model decomposition of the practice effect. Psychon. Bull. Rev. 16(6), 1026–1036 (2009)

    Google Scholar 

  • Eckhoff, P., Wong-Lin, K.F., Holmes, P.: Optimality and robustness of a biophysical decision-making model under nonepinephrine modulation. J. Neurosci. 29(13), 4301–4311 (2009)

    Google Scholar 

  • Eckhoff, P., Wong-Lin, K.F., Holmes, P.: Dimension reduction and dynamics of a spiking neuron model for decision making under neuromodulation. SIAM J. Appl. Dyn. Syst. 10(1), 148–188 (2011)

    MATH  MathSciNet  Google Scholar 

  • Ermentrout, G.B., Kopell, N.: Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J. Appl. Math. 46, 233–253 (1986)

    MATH  MathSciNet  Google Scholar 

  • Ermentrout, G.B., Terman, D.: Mathematical Foundations of Neuroscience. Springer, New York (2010)

    MATH  Google Scholar 

  • Feng, S., Holmes, P., Rorie, A., Newsome, W.T.: Can monkeys choose optimally when faced with noisy stimuli and unequal rewards? PLoS Comput. Biol. 5(2), e1000284 (2009)

    MathSciNet  Google Scholar 

  • FitzHugh, R.: Impulses and physiological states in models of nerve membrane. Biophys. J. 1, 445–466 (1961)

    Google Scholar 

  • Foucaud-Trocme, N., Brunel, N.: Dynamics of the instantaneous firing rate in response to changes in input statistics. J. Comput. Neurosci. 18, 311–321 (2005)

    MathSciNet  Google Scholar 

  • Foucaud-Trocme, N., Hansel, D., van Vreeswijk, C., Brunel, N.: How spike generation mechanisms determine the neuronal response to fluctuating inputs. J. Neurosci. 23, 11628–11640 (2003)

    Google Scholar 

  • Fries, P.: A mechanism for cognitive dynamics: neuronal communication through neuronal coherence. Trends Cogn. Sci. 9, 474–480 (2005)

    Google Scholar 

  • Fries, P., Nikolic, D., Singer, W.: The gamma cycle. Trends Neurosci. 30, 309–316 (2007)

    Google Scholar 

  • Fuchs, E., Holmes, P., Kiemel, T., Ayali, A.: Intersegmental coordination of cockroach locomotion: adaptive control of centrally coupled pattern generator circuits. Frontiers Neural Circuits 4, 125 (2011)

    Google Scholar 

  • Gabbiani, F., Cox, S.: Mathematics for Neuroscientists. Academic Press, San Diego (2010)

    Google Scholar 

  • Gao, J., Wong-Lin, K.F., Holmes, P., Simen, P., Cohen, J.D.: Sequential effects in two-choice reaction time tasks: decomposition and synthesis of mechanisms. Neural Comput. 21(9), 2407–2436 (2009)

    MATH  MathSciNet  Google Scholar 

  • Gao, J., Tortell, R., McClelland, J.L.: Dynamic integration of reward and stimulus information in perceptual decision-making. PLoS ONE 6(3), e16749 (2011)

    Google Scholar 

  • Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, New York (1985), 3rd edn. (2004)

    Google Scholar 

  • Gervasoni, D., Lin, S.-C., Ribeiro, S., Soares, E.S., Pantoja, J., Nicolelis, M.A.L.: Global forebrain dynamics predict rat behavioral states and their transitions. J. Neurosci. 24(49), 11137–11147 (2004)

    Google Scholar 

  • Getting, P.A.: Comparative analysis of invertebrate central pattern generators. In: Cohen, A.H., Rossignol, S., Grillner, S. (eds.) Neural Control of Rhythmic Movements in Vertebrates, pp. 101–128. Wiley, New York (1988), Chap. 4

    Google Scholar 

  • Ghigliazza, R.M., Holmes, P.: A minimal model of a central pattern generator and motoneurons for insect locomotion. SIAM J. Appl. Dyn. Syst. 3(4), 671–700 (2004a)

    MATH  MathSciNet  Google Scholar 

  • Ghigliazza, R.M., Holmes, P.: Minimal models of bursting neurons: how multiple currents, conductances and timescales affect bifurcation diagrams. SIAM J. Appl. Dyn. Syst. 3(4), 636–670 (2004b)

    MATH  MathSciNet  Google Scholar 

  • Gibbon, J.: Scalar expectancy theory and Weber’s law in animal timing. Psychol. Rev. 84(3), 279–325 (1977)

    Google Scholar 

  • Gold, J.I., Shadlen, M.N.: Neural computations that underlie decisions about sensory stimuli. Trends Cogn. Sci. 5(1), 10–16 (2001)

    Google Scholar 

  • Gold, J., Shadlen, M.: Banburismus and the brain: decoding the relationship between sensory stimuli, decisions, and reward. Neuron 36, 299–308 (2002)

    Google Scholar 

  • Gold, J.I., Shadlen, M.N.: Banburismus and the brain: decoding the relationship between sensory stimuli, decisions, and reward. Neuron 36, 299–308 (2002)

    Google Scholar 

  • Goldfarb, S., Wong-Lin, K.F., Schwemmer, M., Leonard, N.E., Holmes, P.: Can post-error dynamics explain sequential reaction time patterns? Frontiers Psychol. 3, 213 (2012)

    Google Scholar 

  • Gray, C.M., Singer, W.: Visual feature integration and the temporal correlation hypothesis. Annu. Rev. Neurosci. 18, 555–586 (1995)

    Google Scholar 

  • Gray, C.M., König, P., Engel, A.K., Singer, W.: Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338, 334–337 (1989)

    Google Scholar 

  • Grillner, S.: Bridging the gap—from ion channels to networks and behaviour. Curr. Opin. Neurobiol. 9, 663–669 (1999)

    Google Scholar 

  • Grossberg, S.: Nonlinear neural networks: principles, mechanisms, and architectures. Neural Netw. 1, 17–61 (1988)

    Google Scholar 

  • Guckenheimer, J.: Isochrons and phaseless sets. J. Math. Biol. 1, 259–273 (1975)

    MATH  MathSciNet  Google Scholar 

  • Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983), 6th edn. (2002)

    MATH  Google Scholar 

  • Guckenheimer, J., Johnson, S.: Planar hybrid systems. In: Lecture Notes in Computer Science, vol. 999, pp. 202–225. Springer, Berlin (1995)

    Google Scholar 

  • Hagevik, A., McClellan, A.D.: Coupling of spinal locomotor networks in larval lamprey revealed by receptor blockers for inhibitory amino acids: neurophysiology and computer modeling. J. Neurophysiol. 72, 1810–1829 (1994)

    Google Scholar 

  • Hellgren, J., Grillner, S., Lansner, A.: Computer simulation of the segmental neural network generating locomotion in lamprey by using populations of network interneurons. Biol. Cybern. 68, 1–13 (1992)

    Google Scholar 

  • Hill, A.A., Lu, J., Masino, M.A., Olsen, O.H., Calabrese, R.L.: A model of a segmental oscillator in the leech heartbeat neuronal network. J. Comput. Neurosci. 10, 281–302 (2001)

    Google Scholar 

  • Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds. Springer Lecture Notes in Mathematics, vol. 583. Springer, Berlin (1977)

    MATH  Google Scholar 

  • Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems and an Introduction to Chaos. Academic Press/Elsevier, San Diego (2004)

    MATH  Google Scholar 

  • Hodgkin, A.L., Huxley, A.F.: The components of membrane conductance in the giant axon of Loligo. J. Physiol. 116, 473–496 (1952a)

    Google Scholar 

  • Hodgkin, A.L., Huxley, A.F.: Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J. Physiol. 116, 449–472 (1952b)

    Google Scholar 

  • Hodgkin, A.L., Huxley, A.F.: The dual effect of membrane potential on sodium conductance in the giant axon of Loligo. J. Physiol. 116, 497–506 (1952c)

    Google Scholar 

  • 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 (1952d)

    Google Scholar 

  • Hodgkin, A.L., Huxley, A.F., Katz, B.: Ionic currents underlying activity in the giant axon of the squid. Arch. Sci. Physiol. 3, 129–150 (1949)

    Google Scholar 

  • Hodgkin, A.L., Huxley, A.F., Katz, B.: Measurement of current-voltage relations in the membrane of the giant axon of Loligo. J. Physiol. 116, 424–448 (1952)

    Google Scholar 

  • Holmes, M.H.: Introduction to Perturbation Methods, 2nd edn. Springer, New York (2013)

    MATH  Google Scholar 

  • Holmes, P., Cohen, J.D.: Optimality and some of its discontents: successes and shortcomings of existing models for binary decisions. Top. Cog. Sci. 6 (2014). To appear

  • Holmes, P., Full, R.J., Koditschek, D., Guckenheimer, J.: The dynamics of legged locomotion: models, analyses and challenges. SIAM Rev. 48(2), 207–304 (2006)

    MATH  MathSciNet  Google Scholar 

  • Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks. Springer, New York (1997)

    Google Scholar 

  • Ijspeert, A.J.: Central pattern generators for locomotion control in animals and robots: a review. Neural Netw. 21, 642–653 (2008)

    Google Scholar 

  • Insel, T.R., Landis, S.C., Collins, F.S.: The NIH BRAIN initiative. Science 340(6133), 687–688 (2013)

    Google Scholar 

  • Izhikevich, E.M.: Which model to use for cortical spiking neurons? IEEE Trans. Neural Netw. 15, 1063–1070 (2004)

    Google Scholar 

  • Johnston, D., Wu, S.: Foundations of Cellular Neurophysiology. MIT Press, Cambridge (1997)

    Google Scholar 

  • Jones, C.K.R.T.: Geometric Singular Perturbation Theory. Lecture Notes in Mathematics, vol. 1609. Springer, Heidelberg (1994). C.I.M.E. Lectures

    Google Scholar 

  • Jones, A.D., Cho, R.Y., Nystrom, L.E., Cohen, J.D., Brown, E.T., Braver, T.S.: A computational model of anterior cingulate function in speeded response tasks: effects of frequency, sequence, and conflict. Cog. Affect. Behav. Neurosci. 2(4), 300–317 (2002)

    Google Scholar 

  • Jones, L.M., Fontanini, A., Sadacca, B.F., Miller, P., Katz, D.B.: Natural stimuli evoke dynamic sequences of states in sensory cortical ensembles. Proc. Natl. Acad. Sci. USA 104(47), 18772–18777 (2007)

    Google Scholar 

  • Keener, J., Sneyd, J.: Mathematical Physiology, 2nd edn. Springer, New York (2009), 2 vols

    Google Scholar 

  • Knight, B.W.: Dynamics of encoding in a population of neurons. J. Gen. Physiol. 59, 734–766 (1972a)

    Google Scholar 

  • Knight, B.W.: The relationship between the firing rate of a single neuron and the level of activity in a population of neurons. Experimental evidence for resonant enhancement in the population response. J. Gen. Physiol. 59, 767–778 (1972b)

    Google Scholar 

  • Knill, D.C., Pouget, A.: The Bayesian brain: the role of uncertainty in neural coding and computation. Trends Neurosci. 27(12) (2004)

  • Kopell, N.: Toward a theory of modelling central pattern generators. In: Cohen, A.H., Rossignol, S., Grillner, S. (eds.) Neural Control of Rhythmic Movements in Vertebrates, pp. 3369–3413. Wiley, New York (1988)

    Google Scholar 

  • Kopell, N., Traub, R.D., Whittington, M.A.: Modeling rhythms: from physiology to function. In: Short Course 2, Rhythms of the Neocortex: Where Do They Come From and What Are They Good For? Society for Neuroscience (2009). http://www.sfn.org/shortcourses

  • Kopell, N., Börgers, C., Pervouchine, D., Malerba, P., Tort, A.B.L.: Gamma and theta rhythms in biophysical models of hippocampal circuits. In: Cutsuridis, V., Graham, B.F., Cobb, S., Vida, I. (eds.) Hippocampal Microcircuits: A Computational Modeller’s Resource Book, pp. 423–457. Springer, New York (2010)

    Google Scholar 

  • Kording, K.P., Wolpert, D.M.: Bayesian decision theory in sensorimotor control. Trends Cogn. Sci. 10, 319–326 (2006)

    Google Scholar 

  • Kording, K.P., Ku, S., Wolpert, D.: Bayesian integration in force estimation. J. Neurophysiol. 92, 3161–3165 (2004)

    Google Scholar 

  • Krinsky, V.I., Kokoz, Yu.M.: Reduction of the Hodgkin-Huxley system to a second order system. Biofizika 18(3), 506–511 (1973)

    Google Scholar 

  • Kukillaya, R.P., Holmes, P.: A hexapedal jointed-leg model for insect locomotion in the horizontal plane. Biol. Cybern. 97, 379–395 (2007)

    MATH  MathSciNet  Google Scholar 

  • Kukillaya, R.P., Holmes, P.: A model for insect locomotion in the horizontal plane: feedforward activation of fast muscles, stability, and robustness. J. Theor. Biol. 261(2), 210–226 (2009)

    MathSciNet  Google Scholar 

  • Kukillaya, R.P., Proctor, J., Holmes, P.: Neuro-mechanical models for insect locomotion: Stability, maneuverability, and proprioceptive feedback. CHAOS: Interdiscip. J. Nonlinear Sci. 19(2), 026107 (2009)

    MathSciNet  Google Scholar 

  • Laming, D.R.J.: Information Theory of Choice-Reaction Times. Academic Press, New York (1968)

    Google Scholar 

  • Lapicque, L.: Recherches quantitatives sur l’excitatabilité électrique des nerfs traitée comme une polarisation. J. Physiol. Pathol. Gen. 9, 620–635 (1907)

    Google Scholar 

  • Latham, P., Brunel, N.: Firing rate of the noisy quadratic integrate-and-fire neuron. Neural Comput. 15, 2281–2306 (2003)

    MATH  Google Scholar 

  • Latham, P.E., Richmond, B.J., Nelson, P.G., Nirenberg, S.: Intrinsic dynamics in neuronal networks. I. Theory. J. Neurophysiol. 83, 808–827 (2000)

    Google Scholar 

  • Liu, Y.S., Holmes, P., Cohen, J.D.: A neural network model of the Eriksen task: reduction, analysis, and data fitting. Neural Comput. 20(2), 345–373 (2008)

    MATH  MathSciNet  Google Scholar 

  • Liu, Y.S., Yu, A.J., Holmes, P.: Dynamical analysis of Bayesian inference models for the Eriksen task. Neural Comput. 21(6), 1520–1553 (2009)

    MATH  MathSciNet  Google Scholar 

  • London, M., Häusser, M.: Dendritic computation. Annu. Rev. Neurosci. 28, 503–532 (2005)

    Google Scholar 

  • Ma, W.J., Beck, J.M., Latham, P.E., Pouget, A.: Bayesian inference with probabilistic population codes. Nat. Neurosci. 9, 1432–1438 (2006)

    Google Scholar 

  • Ma, W.J., Beck, J.M., Pouget, A.: Spiking networks for Bayesian inference and choice. Curr. Opin. Neurobiol. 18, 217–222 (2008)

    Google Scholar 

  • MacKay, D.J.C.: Information Theory, Inference and Learning Algorithms. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  • Mackey, M.C., Santillán, M.: Andrew Fielding Huxley (1917–1952). AMS Not. 60(5), 576–584 (2013)

    Google Scholar 

  • Maddox, W.T., Bohil, C.J.: Base-rate and payoff effects in multidimensional perceptual categorization. J. Exp. Psychol. 24(6), 1459–1482 (1998)

    Google Scholar 

  • Malkin, I.G.: Methods of Poincaré and Linstedt in the Theory of Nonlinear Oscillations. Gostexisdat, Moscow (1949). In Russian

    Google Scholar 

  • Malkin, I.G.: Some Problems in Nonlinear Oscillation Theory. Gostexisdat, Moscow (1956). In Russian

    Google Scholar 

  • Marder, E.: Motor pattern generation. Curr. Opin. Neurobiol. 10(6), 691–698 (2000)

    Google Scholar 

  • Marder, E., Bucher, D.: Understanding circuit dynamics using the stomatogastric nervous system of lobsters and crabs. Annu. Rev. Physiol. 69, 291–316 (2007)

    Google Scholar 

  • Markram, H.: The blue brain project. Nat. Rev. Neurosci. 7, 153–160 (2006)

    Google Scholar 

  • Mazurek, M.E., Roitman, J.D., Ditterich, J., Shadlen, M.N.: A role for neural integrators in perceptual decision making. Cereb. Cortex 13(11), 891–898 (2003)

    Google Scholar 

  • McCarthy, M.M., Moore-Kochlacs, C., Gu, X., Boyden, E.S., Han, X., Kopell, N.: Striatal origin of the pathologic beta oscillations in Parkinson’s disease. Proc. Nat. Acad. Sci. 108(28), 11620–11625 (2011)

    Google Scholar 

  • McCarthy, M.M., Ching, S., Whittington, M.A., Kopell, N.: Dynamical changes in neurological disease and anesthesia. Curr. Opin. Neurobiol. 22(4), 693–703 (2012)

    Google Scholar 

  • McGinn, C.: What can your neurons tell you? New York Review of Books. LX 12:49–50 (2013), by J.-P. Changeux

    Google Scholar 

  • McMillen, T., Holmes, P.: An elastic rod model for anguilliform swimming. J. Math. Biol. 53, 843–866 (2006)

    MathSciNet  Google Scholar 

  • McMillen, T., Williams, T.L., Holmes, P.: Nonlinear muscles, passive viscoelasticity and body taper conspire to create neuro-mechanical phase lags in anguilliform swimmers. PLoS Comput. Biol. 4(8), e1000157 (2008)

    MathSciNet  Google Scholar 

  • Miller, K.D., Fumarola, F.: Mathematical equivalence of two common forms of firing rate models of neural networks. Neural Comput. 24, 25–31 (2012)

    MATH  MathSciNet  Google Scholar 

  • Myung, J., Busemeyer, J.R.: Criterion learning in a deferred decision-making task. Am. J. Psychol. 102(1), 1–16 (1989)

    Google Scholar 

  • Nagumo, J.S., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating a nerve axon. Proc. IRE 50, 2061–2070 (1962)

    Google Scholar 

  • Pearson, K.G.: Central programming and reflex control of walking in the cockroach. J. Exp. Biol. 56, 173–193 (1972)

    Google Scholar 

  • Pearson, K.G.: Motor systems. Curr. Opin. Neurobiol. 10, 649–654 (2000)

    Google Scholar 

  • Pearson, K.G., Fourtner, C.R.: Non-spiking interneurons in walking system of the cockroach. J. Neurophysiol. 38, 33–51 (1975)

    Google Scholar 

  • Pearson, K.G., Iles, J.F.: Discharge patterns of coxal levator and depressor motoneurones of the cockroach, Periplaneta americana. J. Exp. Biol. 52, 139–165 (1970)

    Google Scholar 

  • Pearson, K.G., Iles, J.F.: Nervous mechanisms underlying intersegmental co-ordination of leg movements during walking in the cockroach. J. Exp. Biol. 58, 725–744 (1973)

    Google Scholar 

  • Petrov, A.A., Van Horn, N.M., Ratcliff, R.: Dissociable perceptual-learning mechanisms revealed by diffusion-model analysis. Psychon. Bull. Rev. 18(3), 490–497 (2011)

    Google Scholar 

  • Posner, M.I., Snyder, C.R.R.: Attention and cognitive control. In: Solso, R. (ed.) Information Processing and Cognition: The Loyola Symposium. Lawrence Erlbaum, Hillsdale (1975)

    Google Scholar 

  • Pouget, A., Dayan, P., Zemel, R.S.: Inference and computation with population codes. Annu. Rev. Neurosci. 26, 381–410 (2003)

    Google Scholar 

  • Proctor, J., Holmes, P.: Reflexes and preflexes: on the role of sensory feedback on rhythmic patterns in legged locomotion. Biol. Cybern. 2, 513–531 (2010)

    MathSciNet  Google Scholar 

  • Rall, W.: Branching dendritic trees and motoneuron membrane resistivity. Exp. Neurol. 1(5), 491–517 (1959)

    Google Scholar 

  • Rall, W.: Theoretical significance of dendritic trees for neuronal input-output relations. In: Reiss, R. (ed.) Neural Theory and Modeling, pp. 93–97. Stanford Univ. Press, Stanford (1964)

    Google Scholar 

  • Rall, W., Burke, R.E., Smith, T.G., Nelson, P.G., Frank, K.: Dendritic location of synapses and possible mechanisms for the monosynaptic EPSP in motoneurons. J. Neurophysiol. 30, 1169–1193 (1967)

    Google Scholar 

  • Ratcliff, R.: A theory of memory retrieval. Psychol. Rev. 85, 59–108 (1978)

    Google Scholar 

  • Ratcliff, R., Van Zandt, T., McKoon, G.: Connectionist and diffusion models of reaction time. Psychol. Rev. 106(2), 261–300 (1999)

    Google Scholar 

  • Ratcliff, R., Cherian, A., Segraves, M.A.: A comparison of macaque behavior and superior colliculus neuronal activity to predictions from models of two choice decisions. J. Neurophysiol. 90, 1392–1407 (2003)

    Google Scholar 

  • Ratcliff, R., Hasegawa, Y.T., Hasegawa, R.P., Smith, P.L., Segraves, M.A.: Dual-diffusion model for single-cell recording data from the superior colliculus in a brightness-discrimination task. J. Neurophysiol. 97, 1756–1774 (2006)

    Google Scholar 

  • Renart, A., Brunel, N., Wang, X.-J.: Mean-field theory of recurrent cortical networks: from irregularly spiking neurons to working memory. In: Feng, J. (ed.) Computational Neuroscience: A Comprehensive Approach. CRC Press, Boca Raton (2003)

    Google Scholar 

  • Rieke, F., Bialek, W., Warland, D., De Ruyter van Steveninck, R.: Spikes: Exploring the Neural Code. MIT Press, Cambridge (1997)

    Google Scholar 

  • Rinzel, J.: Excitation dynamics: insights from simplified membrane models. Fed. Proc. 37, 2944–2946 (1985)

    Google Scholar 

  • Roitman, J., Shadlen, M.: Response of neurons in the lateral intraparietal area during a combined visual discrimination reaction time task. J. Neurosci. 22, 9475–9489 (2002)

    Google Scholar 

  • Rorie, A., Gao, J., McClelland, J.L., Newsome, W.T.: Integration of sensory and reward information during perceptual decision-making in lateral intraparietal cortex (LIP). PLoS ONE 5(2), e9308 (2010)

    Google Scholar 

  • Roxin, A., Ledberg, A.: Neurobiological models of two-choice decision making can be reduced to a one-dimensional nonlinear diffusion equation. PLoS Comput. Biol. 4(3), e10000146 (2008)

    MathSciNet  Google Scholar 

  • Rumelhart, D.E., McClelland, J.L.: Parallel Distributed Processing: Explorations in the Microstructure of Cognition. MIT Press, Cambridge (1986)

    Google Scholar 

  • Sara, S.J.: The locus coeruleus and noradrenergic modulation of cognition. Nat. Rev. Neurosci. 10, 211–223 (2009)

    Google Scholar 

  • Schall, J.D.: Neural basis of deciding, choosing and acting. Nat. Rev. Neurosci. 2, 33–42 (2001)

    Google Scholar 

  • Schultz, W., Dayan, P., Montague, P.R.: A neural substrate of prediction and reward. Science 275, 1593–1599 (1997)

    Google Scholar 

  • Schultz, W., Tremblay, K.L., Hollerman, J.R.: Reward processing in primate orbitofrontal cortex and basal ganglia. Cereb. Cortex 10, 272–284 (2000)

    Google Scholar 

  • Seidemann, E., Meilijson, I., Abeles, M., Bergman, H., Vaadia, E.: Simultaneously recorded single units in the frontal cortex go through sequences of discrete and stable states in monkeys performing a delayed localization task. J. Neurosci. 16(2), 752–768 (1996)

    Google Scholar 

  • Servan-Schreiber, D., Printz, H., Cohen, J.D.: A network model of catecholamine effects: gain, signal-to-noise ratio, and behavior. Science 249, 892–895 (1990)

    Google Scholar 

  • Seung, S.: Connectome: How the Brain’s Wiring Makes Us Who We Are. Houghton Mifflin Harcourt, New York (2012)

    Google Scholar 

  • Shadlen, M.N., Newsome, W.T.: Neural basis of a perceptual decision in the parietal cortex (area LIP) of the rhesus monkey. J. Neurophysiol. 86, 1916–1936 (2001)

    Google Scholar 

  • Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948), also pp. 623–656. Reprinted in (Knill and Pouget 2004; Liu et al. 2009; Yu et al. 2009; Bejjanki et al. 2011; Dayan 2012; Solway and Botvinick 2012)

    MATH  MathSciNet  Google Scholar 

  • Shannon, C.E., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press, Urbana (1949)

    MATH  Google Scholar 

  • Sherman, A., Rinzel, J., Keizer, J.: Emergence of organized bursting in clusters of pancreatic β-cells by channel sharing. Biophys. J. 54, 411–425 (1988)

    Google Scholar 

  • Simen, P., Contreras, D., Buck, C., Hu, P., Holmes, P., Cohen, J.D.: Reward rate optimization in two-alternative decision making: empirical tests of theoretical predictions. J. Exp. Psychol. Hum. Percept. Perform. 35(6), 1865–1897 (2009)

    Google Scholar 

  • Smith, P.L., Ratcliff, R.: Psychology and neurobiology of simple decisions. Trends Neurosci. 27(3), 161–168 (2004)

    Google Scholar 

  • Soetens, E., Boer, L., Hueting, J.: Expectancy or automatic facilitation? Separating sequential effects in two-choice reaction times. J. Exp. Psychol. Hum. Percept. Perform. 11, 598–616 (1985)

    Google Scholar 

  • Solway, A., Botvinick, M.M.: Goal-directed decision making as probabilistic inference: a computational framework and potential neural correlates. Psychol. Rev. 119, 120–154 (2012)

    Google Scholar 

  • Stein, R.: A theoretical analysis of neuronal variability. Biophys. J. 5, 173–194 (1965)

    Google Scholar 

  • Stone, M.: Models for choice-reaction time. Psychometrika 25, 251–260 (1960)

    MATH  Google Scholar 

  • Sutton, R.: Learning to predict by the method of temporal differences. Mach. Learn. 3, 9–44 (1988)

    Google Scholar 

  • Sutton, R.S., Barto, A.G.: Reinforcement Learning. MIT Press, Cambridge (1998)

    Google Scholar 

  • Swinehard, C., Abbott, L.F.: Dimensional reduction in reward-based learning. Netw. Comput. Neural Syst. 17, 235–252 (2006)

    Google Scholar 

  • Torta, A.B.L., Komorowskie, R.W., Manns, J.R., Kopell, N.J., Eichenbaum, H.: Theta-gamma coupling increases during the learning of item-context associations. Proc. Natl. Acad. Sci. USA 106(49), 20942–20947 (2009)

    Google Scholar 

  • Toth, T.I., Daun-Gruhn, S.: A putative neuronal network controlling the activity of the leg motoneurons of the stick insect. NeuroReport 22(18), 943–946 (2011)

    Google Scholar 

  • Tytell, E.D., Hsu, C.-Y., Williams, T.L., Cohen, A.H., Fauci, L.J.: Interactions between body stiffness, muscle activation, and fluid environment in a neuromechanical model of lamprey swimming. Proc. Natl. Acad. Sci. 107(46), 19832–19837 (2010)

    Google Scholar 

  • Tytell, E.D., Holmes, P., Cohen, A.H.: Spikes alone do not behavior make: why neuroscience needs biomechanics. Curr. Opin. Neurobiol. 21, 816–822 (2011)

    Google Scholar 

  • Uhlhaas, P.J., Singer, W.: Abnormal neural oscillations and synchrony in schizophrenia. Nat. Rev. Neurosci. 11, 100–113 (2010)

    Google Scholar 

  • Usher, M., Cohen, J.D.: Short term memory and selection processes in a frontal-lobe model. In: Heinke, D., Humphreys, G.W., Olson, A.J. (eds.) Connectionist models in cognitive neuroscience: the 5th neural computation and psychology workshop, pp. 78–91. Springer, Birmingham (1999)

    Google Scholar 

  • Usher, M., McClelland, J.L.: On the time course of perceptual choice: the leaky competing accumulator model. Psychol. Rev. 108, 550–592 (2001)

    Google Scholar 

  • van Ravenzwaaij, D., van der Maas, H.L.J., Wagenmakers, E.J.: Optimal decision making in neural inhibition models. Psychol. Rev. 119(1), 201–215 (2012)

    Google Scholar 

  • Várkonyi, P.L., Kiemel, T., Hoffman, K., Cohen, A.H., Holmes, P.: On the derivation and tuning of phase oscillator models for lamprey central pattern generators. J. Comput. Neurosci. 25, 245–261 (2008)

    MathSciNet  Google Scholar 

  • von Neumann, J.: The Computer and the Brain, 2nd edn. Yale University Press, New Haven (1958). With a foreward by P.M. and Churchland P.S. (2000)

    MATH  Google Scholar 

  • Wald, A., Wolfowitz, J.: Optimum character of the sequential probability ratio test. Ann. Math. Stat. 19, 326–339 (1948)

    MATH  MathSciNet  Google Scholar 

  • Wallen, P., Ekeberg, O., Lansner, A., Brodin, L., Traven, H., Grillner, S.: A computer-based model for realistic simulations of neural networks. II. The segmental network generating locomotor rhythmicity in the lamprey. J. Neurophysiol. 68, 1939–1950 (1992)

    Google Scholar 

  • Wang, X.-J.: Synaptic basis of cortical persistent activity: the importance of NMDA receptors to working memory. J. Neurosci. 19, 9587–9603 (1999)

    Google Scholar 

  • Wang, X.-J.: Probabilistic decision making by slow reverberation in cortical circuits. Neuron 36, 955–968 (2002)

    Google Scholar 

  • Wang, X.-J.: Decision making in recurrent neuronal circuits. Neuron 60, 215–234 (2008)

    Google Scholar 

  • Wang, X.-J.: Neurophysiological and computational principles of cortical rhythms in cognition. Physiol. Rev. 90, 1195–1268 (2010)

    Google Scholar 

  • Whittington, M.A., Traub, R.D., Kopell, N.: Diverse origins of network rhythms in cortical local circuits. In: Short Course 2, Rhythms of the Neocortex: Where do They Come From and What Are They Good For? Society for Neuroscience (2009). http://www.sfn.org/shortcourses

  • Wiener, N.: Cybernetics: Or Control and Communication in the Animal and the Machine, 2nd edn. MIT Press, Cambridge (1948). 1961

    Google Scholar 

  • Wilson, H.: Spikes, Decisions and Actions: The Dynamical Foundations of Neuroscience. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  • Winfree, A.T.: The Geometry of Biological Time, 2nd edn. Springer, New York (2001)

    MATH  Google Scholar 

  • Wolpert, D.: Probabilistic models in human sensorimotor control. Hum. Mov. Sci. 26, 511–524 (2007)

    Google Scholar 

  • Wong, K.F., Wang, X.-J.: A recurrent network mechanism of time integration in perceptual decisions. J. Neurosci. 26(4), 1314–1328 (2006)

    Google Scholar 

  • Yu, A.J., Cohen, J.D.: Sequential effects: superstition or rational behavior? In: Platt, J.C., Koller, D., Singer, Y., Roweis, S. (eds.) Advances in Neural Information Processing Systems, vol. 21, pp. 1873–1880. MIT Press, Cambridge (2009)

    Google Scholar 

  • Yu, A.J., Dayan, P., Cohen, J.D.: Dynamics of attentional selection under conflict: toward a rational Bayesian account. J. Exp. Psychol. Hum. Percept. Perform. 35(3), 700–717 (2009)

    Google Scholar 

  • Zacksenhouse, M., Holmes, P., Bogacz, R.: Robust versus optimal strategies for two-alternative forced choice tasks. J. Math. Psychol. 54, 230–246 (2010)

    MATH  MathSciNet  Google Scholar 

  • Zhang, M., Barash, S.: Persistent LIP activity in memory antisaccades: working memory for a sensorimotor transformation. J. Neurophysiol. 91, 1424–1441 (2004)

    Google Scholar 

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Acknowledgements

This article draws upon work done since 2000, variously supported by NSF under DMS-0101208 and EF-0425878, AFOSR under FA9550-07-1-0537 and FA9550-07-1-0528, NIH under P50 MH62196, the US-Israel Binational Science Foundation under BSF 2011059, and the J. Insley Pyne Fund of Princeton University. The material in Sects. 2–3 was adapted from notes for a course taught at Princeton since 2006 to which Philip Eckhoff contributed much. An extended version of Sect. 4 appears in Kording et al. (2004), Kording and Wolpert (2006), Wolpert (2007). The author thanks Fuat Balci for providing Fig. 12 and the anonymous reviewers for their suggestions, and gratefully acknowledges the contributions of many other collaborators, not all of whose work could be cited here.

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  1. Department of Mechanical and Aerospace Engineering, Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Philip Holmes

  2. Princeton Neuroscience Institute, Princeton University, Princeton, NJ, 08544, USA

    Philip Holmes

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Holmes, P. Some Joys and Trials of Mathematical Neuroscience. J Nonlinear Sci 24, 201–242 (2014). https://doi.org/10.1007/s00332-013-9191-4

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