# American Institute of Mathematical Sciences

September  2016, 21(7): 2255-2273. doi: 10.3934/dcdsb.2016046

## Neurotransmitter concentrations in the presence of neural switching in one dimension

 1 Department of Mathematics, University of Utah, Salt Lake City, UT 84112, United States 2 Department of Mathematics, The Ohio State University, Columbus, OH 43210, United States 3 Department of Mathematics, Duke University, Durham, NC 27708, United States

Received  September 2015 Revised  March 2016 Published  August 2016

In volume transmission, neurons in one brain nucleus send their axons to a second nucleus where neurotransmitter is released into the extracellular space. One would like methods to calculate the average amount of neurotransmitter at different parts of the extracellular space, depending on neural properties and the geometry of the projections and the extracellular space. This question is interesting mathematically because the neuron terminals are both the sources (when they are firing) and the sinks (when they are quiescent) of neurotransmitter. We show how to formulate the questions as boundary value problems for the heat equation with stochastically switching boundary conditions. In one space dimension, we derive explicit formulas for the average concentration in terms of the parameters of the problems in two simple prototype examples and then explain how the same methods can be used to solve the general problem. Applications of the mathematical results to the neuroscience context are discussed.
Citation: Sean D. Lawley, Janet A. Best, Michael C. Reed. Neurotransmitter concentrations in the presence of neural switching in one dimension. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2255-2273. doi: 10.3934/dcdsb.2016046
##### References:
 [1] C. W. Atcherley, K. M. Wood, K. L. Parent, P. Hashemi and M. L. Heien, The coaction of tonic and phasic dopamine dynamics, Chem. Commun., 51 (2015), 2235-2238. doi: 10.1039/C4CC06165A. [2] P. Blandina, J. Goldfarb, B. Craddock-Royal and J. P. Green, Release of endogenous dopamine by stimulation of 5-hydroxytryptamine3 receptors in rat striatum, J. Pharmacol. Exper. Therap., 251 (1989), 803-809. [3] N. Bonhomme, P. Duerwaerdere, M. Moal and U. Spampinato, Evidence for 5-HT4 receptor subtype involvement in the enhancement of striatal dopamine release induced by serotonin: A microdialysis study in the halothane-anesthetized rat, Neuropharmacology, 34 (1995), 269-279. doi: 10.1016/0028-3908(94)00145-I. [4] P. C. Bressloff and S. D. Lawley, Escape from a potential well with a randomly switching boundary, J. Phys. A, 48 (2015), 225001, 25pp, URL http://dx.doi.org/10.1088/1751-8113/48/22/225001. doi: 10.1088/1751-8113/48/22/225001. [5] P. C. Bressloff and S. D. Lawley, Escape from subcellular domains with randomly switching boundaries, Multiscale Model. Simul., 13 (2015), 1420-1445, URL http://dx.doi.org/10.1137/15M1019258. doi: 10.1137/15M1019258. [6] D. J. Brooks, Dopamine agonists: their role in the treatment of Parkinson's disease, J. Neurol. Neurosurg Psychiatry, 68 (2000), 685-689. doi: 10.1136/jnnp.68.6.685. [7] H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc. (2), 63 (2001), 413-427, URL http://dx.doi.org/10.1017/S0024610700001915. doi: 10.1017/S0024610700001915. [8] L. C. Daws, W. Koek and N. C. Mitchell, Revisiting serotonin reuptake inhibitors and the therapeutic effects of "uptake 2'' in psychiatric disorders, ACS Chem. Neurosci., 4 (2013), 16-21. [9] L. Daws, S. Montenez, W. Owens, G. Gould, A. Frazer, G. Toney and G. Gerhardt, Transport mechanisms governing serotonin clearance in vivo revealed by high speed chronoamperometry, J Neurosci Meth, 143 (2005), 49-62. doi: 10.1016/j.jneumeth.2004.09.011. [10] R. Feldman, J. Meyer and L. Quenzer, Principles of Neuropharmacology, Sinauer Associates, Inc, Sunderland, MA., 1997. [11] K. Fuxe, A. B. Dahlstrom, G. Jonsson, D. Marcellino, M. Guescini, M. Dam, P. Manger and L. Agnati, The discovery of central monoamine neurons gave volume transmission to the wired brain, Prog. Neurobiol., 90 (2010), 82-100. doi: 10.1016/j.pneurobio.2009.10.012. [12] M. Hajos, S. E. Gartside, A. E. P. Villa and T. Sharp, Evidence for a repetitive (burst) firing pattern in a sub-population of 5-hydroxytryptamine neurons in the dorsal and median raphe nuclei of the rat, Neuroscience, 69 (1995), 189-197. doi: 10.1016/0306-4522(95)00227-A. [13] E. Kandel, J. Schwartz, T. Jessell, S. Siegelbaum and A. Hudspeth, Principles of Neural Science, 5th edition, McGraw-Hill Education / Medical, 2012. [14] S. D. Lawley, Boundary value problems for statistics of diffusion in a randomly switching environment: PDE and SDE perspectives, SIAM J. Appl. Dyn. Syst., 15 (2016), 1410-1433, URL http://dx.doi.org/10.1137/15M1038426. doi: 10.1137/15M1038426. [15] S. D. Lawley and J. P. Keener, A new derivation of Robin boundary conditions through homogenization of a stochastically switching boundary, SIAM J. Appl. Dyn. Syst., 14 (2015), 1845-1867, URL http://dx.doi.org/10.1137/15M1015182. doi: 10.1137/15M1015182. [16] S. D. Lawley, J. C. Mattingly and M. C. Reed, Stochastic switching in infinite dimensions with applications to random parabolic PDE, SIAM Journal on Mathematical Analysis, 47 (2015), 3035-3063, URL http://dx.doi.org/10.1137/140976716. doi: 10.1137/140976716. [17] J. C. Mattingly, Ergodicity of $2$D Navier-Stokes equations with random forcing and large viscosity, Comm. Math. Phys., 206 (1999), 273-288, URL http://dx.doi.org/10.1007/s002200050706. doi: 10.1007/s002200050706. [18] T. Pasik and P. Pasik, Serotonergic afferents in the monkey neostriatum, Acta Biol Acad Sci Hung, 33 (1982), 277-288. [19] M. Reed, H. F. Nijhout and J. Best, Projecting biochemistry over long distances, Math. Model. Nat. Phenom., 9 (2014), 133-138. doi: 10.1051/mmnp/20149109. [20] B. Schmalfuß, A random fixed point theorem based on Lyapunov exponents, Random Comput. Dynam., 4 (1996), 257-268. [21] J.-J. Soghomonian, G. Doucet and L. Descarries, Serotonin innervation in adult rat neostriatum i. quantified regional distribution, Brain Research, 425 (1987), 85-100. doi: 10.1016/0006-8993(87)90486-0. [22] L. Tao and C. Nicholson, Diffusion of albumins in rat cortical slices and relevance to volume transmission, Neuroscience, 75 (1996), 839-847. doi: 10.1016/0306-4522(96)00303-X. [23] , I. Wolfram Research,, Mathematica, (2012). [24] K. M. Wood, A. Zeqja, H. F. Nijhout, M. C. Reed, J. A. Best and P. Hashemi, Voltametric and mathematical evidence for dual transport mediation of serotonin clearance in vivo, J. Neurochem., 130 (2014), 351-359.

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##### References:
 [1] C. W. Atcherley, K. M. Wood, K. L. Parent, P. Hashemi and M. L. Heien, The coaction of tonic and phasic dopamine dynamics, Chem. Commun., 51 (2015), 2235-2238. doi: 10.1039/C4CC06165A. [2] P. Blandina, J. Goldfarb, B. Craddock-Royal and J. P. Green, Release of endogenous dopamine by stimulation of 5-hydroxytryptamine3 receptors in rat striatum, J. Pharmacol. Exper. Therap., 251 (1989), 803-809. [3] N. Bonhomme, P. Duerwaerdere, M. Moal and U. Spampinato, Evidence for 5-HT4 receptor subtype involvement in the enhancement of striatal dopamine release induced by serotonin: A microdialysis study in the halothane-anesthetized rat, Neuropharmacology, 34 (1995), 269-279. doi: 10.1016/0028-3908(94)00145-I. [4] P. C. Bressloff and S. D. Lawley, Escape from a potential well with a randomly switching boundary, J. Phys. A, 48 (2015), 225001, 25pp, URL http://dx.doi.org/10.1088/1751-8113/48/22/225001. doi: 10.1088/1751-8113/48/22/225001. [5] P. C. Bressloff and S. D. Lawley, Escape from subcellular domains with randomly switching boundaries, Multiscale Model. Simul., 13 (2015), 1420-1445, URL http://dx.doi.org/10.1137/15M1019258. doi: 10.1137/15M1019258. [6] D. J. Brooks, Dopamine agonists: their role in the treatment of Parkinson's disease, J. Neurol. Neurosurg Psychiatry, 68 (2000), 685-689. doi: 10.1136/jnnp.68.6.685. [7] H. Crauel, Random point attractors versus random set attractors, J. London Math. Soc. (2), 63 (2001), 413-427, URL http://dx.doi.org/10.1017/S0024610700001915. doi: 10.1017/S0024610700001915. [8] L. C. Daws, W. Koek and N. C. Mitchell, Revisiting serotonin reuptake inhibitors and the therapeutic effects of "uptake 2'' in psychiatric disorders, ACS Chem. Neurosci., 4 (2013), 16-21. [9] L. Daws, S. Montenez, W. Owens, G. Gould, A. Frazer, G. Toney and G. Gerhardt, Transport mechanisms governing serotonin clearance in vivo revealed by high speed chronoamperometry, J Neurosci Meth, 143 (2005), 49-62. doi: 10.1016/j.jneumeth.2004.09.011. [10] R. Feldman, J. Meyer and L. Quenzer, Principles of Neuropharmacology, Sinauer Associates, Inc, Sunderland, MA., 1997. [11] K. Fuxe, A. B. Dahlstrom, G. Jonsson, D. Marcellino, M. Guescini, M. Dam, P. Manger and L. Agnati, The discovery of central monoamine neurons gave volume transmission to the wired brain, Prog. Neurobiol., 90 (2010), 82-100. doi: 10.1016/j.pneurobio.2009.10.012. [12] M. Hajos, S. E. Gartside, A. E. P. Villa and T. Sharp, Evidence for a repetitive (burst) firing pattern in a sub-population of 5-hydroxytryptamine neurons in the dorsal and median raphe nuclei of the rat, Neuroscience, 69 (1995), 189-197. doi: 10.1016/0306-4522(95)00227-A. [13] E. Kandel, J. Schwartz, T. Jessell, S. Siegelbaum and A. Hudspeth, Principles of Neural Science, 5th edition, McGraw-Hill Education / Medical, 2012. [14] S. D. Lawley, Boundary value problems for statistics of diffusion in a randomly switching environment: PDE and SDE perspectives, SIAM J. Appl. Dyn. Syst., 15 (2016), 1410-1433, URL http://dx.doi.org/10.1137/15M1038426. doi: 10.1137/15M1038426. [15] S. D. Lawley and J. P. Keener, A new derivation of Robin boundary conditions through homogenization of a stochastically switching boundary, SIAM J. Appl. Dyn. Syst., 14 (2015), 1845-1867, URL http://dx.doi.org/10.1137/15M1015182. doi: 10.1137/15M1015182. [16] S. D. Lawley, J. C. Mattingly and M. C. Reed, Stochastic switching in infinite dimensions with applications to random parabolic PDE, SIAM Journal on Mathematical Analysis, 47 (2015), 3035-3063, URL http://dx.doi.org/10.1137/140976716. doi: 10.1137/140976716. [17] J. C. Mattingly, Ergodicity of $2$D Navier-Stokes equations with random forcing and large viscosity, Comm. Math. Phys., 206 (1999), 273-288, URL http://dx.doi.org/10.1007/s002200050706. doi: 10.1007/s002200050706. [18] T. Pasik and P. Pasik, Serotonergic afferents in the monkey neostriatum, Acta Biol Acad Sci Hung, 33 (1982), 277-288. [19] M. Reed, H. F. Nijhout and J. Best, Projecting biochemistry over long distances, Math. Model. Nat. Phenom., 9 (2014), 133-138. doi: 10.1051/mmnp/20149109. [20] B. Schmalfuß, A random fixed point theorem based on Lyapunov exponents, Random Comput. Dynam., 4 (1996), 257-268. [21] J.-J. Soghomonian, G. Doucet and L. Descarries, Serotonin innervation in adult rat neostriatum i. quantified regional distribution, Brain Research, 425 (1987), 85-100. doi: 10.1016/0006-8993(87)90486-0. [22] L. Tao and C. Nicholson, Diffusion of albumins in rat cortical slices and relevance to volume transmission, Neuroscience, 75 (1996), 839-847. doi: 10.1016/0306-4522(96)00303-X. [23] , I. Wolfram Research,, Mathematica, (2012). [24] K. M. Wood, A. Zeqja, H. F. Nijhout, M. C. Reed, J. A. Best and P. Hashemi, Voltametric and mathematical evidence for dual transport mediation of serotonin clearance in vivo, J. Neurochem., 130 (2014), 351-359.
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