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

Neurotransmitter concentrations in the presence of neural switching in one dimension

Sean D. Lawley 1, , Janet A. Best 2, and  Michael C. Reed 3,

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.
Keywords: stochastic switching, Heat equation, volume transmission..
Mathematics Subject Classification: Primary: 35R60, 60H15; Secondary: 92C2.
Citation: Sean D. Lawley, Janet A. Best, Michael C. Reed. Neurotransmitter concentrations in the presence of neural switching in one dimension. Discrete & 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.  doi: 10.1039/C4CC06165A.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/0028-3908(94)00145-I.  Google Scholar

[4]

P. C. Bressloff and S. D. Lawley, Escape from a potential well with a randomly switching boundary,, J. Phys. A, 48 (2015), 1751.  doi: 10.1088/1751-8113/48/22/225001.  Google Scholar

[5]

P. C. Bressloff and S. D. Lawley, Escape from subcellular domains with randomly switching boundaries,, Multiscale Model. Simul., 13 (2015), 1420.  doi: 10.1137/15M1019258.  Google Scholar

[6]

D. J. Brooks, Dopamine agonists: their role in the treatment of Parkinson's disease,, J. Neurol. Neurosurg Psychiatry, 68 (2000), 685.  doi: 10.1136/jnnp.68.6.685.  Google Scholar

[7]

H. Crauel, Random point attractors versus random set attractors,, J. London Math. Soc. (2), 63 (2001), 413.  doi: 10.1017/S0024610700001915.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jneumeth.2004.09.011.  Google Scholar

[10]

R. Feldman, J. Meyer and L. Quenzer, Principles of Neuropharmacology,, Sinauer Associates, (1997).   Google Scholar

[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.  doi: 10.1016/j.pneurobio.2009.10.012.  Google Scholar

[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.  doi: 10.1016/0306-4522(95)00227-A.  Google Scholar

[13]

E. Kandel, J. Schwartz, T. Jessell, S. Siegelbaum and A. Hudspeth, Principles of Neural Science,, 5th edition, (2012).   Google Scholar

[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.  doi: 10.1137/15M1038426.  Google Scholar

[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.  doi: 10.1137/15M1015182.  Google Scholar

[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.  doi: 10.1137/140976716.  Google Scholar

[17]

J. C. Mattingly, Ergodicity of $2$D Navier-Stokes equations with random forcing and large viscosity,, Comm. Math. Phys., 206 (1999), 273.  doi: 10.1007/s002200050706.  Google Scholar

[18]

T. Pasik and P. Pasik, Serotonergic afferents in the monkey neostriatum,, Acta Biol Acad Sci Hung, 33 (1982), 277.   Google Scholar

[19]

M. Reed, H. F. Nijhout and J. Best, Projecting biochemistry over long distances,, Math. Model. Nat. Phenom., 9 (2014), 133.  doi: 10.1051/mmnp/20149109.  Google Scholar

[20]

B. Schmalfuß, A random fixed point theorem based on Lyapunov exponents,, Random Comput. Dynam., 4 (1996), 257.   Google Scholar

[21]

J.-J. Soghomonian, G. Doucet and L. Descarries, Serotonin innervation in adult rat neostriatum i. quantified regional distribution,, Brain Research, 425 (1987), 85.  doi: 10.1016/0006-8993(87)90486-0.  Google Scholar

[22]

L. Tao and C. Nicholson, Diffusion of albumins in rat cortical slices and relevance to volume transmission,, Neuroscience, 75 (1996), 839.  doi: 10.1016/0306-4522(96)00303-X.  Google Scholar

[23]

, I. Wolfram Research,, Mathematica, (2012).   Google Scholar

[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.   Google Scholar

show all references

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.  doi: 10.1039/C4CC06165A.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/0028-3908(94)00145-I.  Google Scholar

[4]

P. C. Bressloff and S. D. Lawley, Escape from a potential well with a randomly switching boundary,, J. Phys. A, 48 (2015), 1751.  doi: 10.1088/1751-8113/48/22/225001.  Google Scholar

[5]

P. C. Bressloff and S. D. Lawley, Escape from subcellular domains with randomly switching boundaries,, Multiscale Model. Simul., 13 (2015), 1420.  doi: 10.1137/15M1019258.  Google Scholar

[6]

D. J. Brooks, Dopamine agonists: their role in the treatment of Parkinson's disease,, J. Neurol. Neurosurg Psychiatry, 68 (2000), 685.  doi: 10.1136/jnnp.68.6.685.  Google Scholar

[7]

H. Crauel, Random point attractors versus random set attractors,, J. London Math. Soc. (2), 63 (2001), 413.  doi: 10.1017/S0024610700001915.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jneumeth.2004.09.011.  Google Scholar

[10]

R. Feldman, J. Meyer and L. Quenzer, Principles of Neuropharmacology,, Sinauer Associates, (1997).   Google Scholar

[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.  doi: 10.1016/j.pneurobio.2009.10.012.  Google Scholar

[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.  doi: 10.1016/0306-4522(95)00227-A.  Google Scholar

[13]

E. Kandel, J. Schwartz, T. Jessell, S. Siegelbaum and A. Hudspeth, Principles of Neural Science,, 5th edition, (2012).   Google Scholar

[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.  doi: 10.1137/15M1038426.  Google Scholar

[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.  doi: 10.1137/15M1015182.  Google Scholar

[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.  doi: 10.1137/140976716.  Google Scholar

[17]

J. C. Mattingly, Ergodicity of $2$D Navier-Stokes equations with random forcing and large viscosity,, Comm. Math. Phys., 206 (1999), 273.  doi: 10.1007/s002200050706.  Google Scholar

[18]

T. Pasik and P. Pasik, Serotonergic afferents in the monkey neostriatum,, Acta Biol Acad Sci Hung, 33 (1982), 277.   Google Scholar

[19]

M. Reed, H. F. Nijhout and J. Best, Projecting biochemistry over long distances,, Math. Model. Nat. Phenom., 9 (2014), 133.  doi: 10.1051/mmnp/20149109.  Google Scholar

[20]

B. Schmalfuß, A random fixed point theorem based on Lyapunov exponents,, Random Comput. Dynam., 4 (1996), 257.   Google Scholar

[21]

J.-J. Soghomonian, G. Doucet and L. Descarries, Serotonin innervation in adult rat neostriatum i. quantified regional distribution,, Brain Research, 425 (1987), 85.  doi: 10.1016/0006-8993(87)90486-0.  Google Scholar

[22]

L. Tao and C. Nicholson, Diffusion of albumins in rat cortical slices and relevance to volume transmission,, Neuroscience, 75 (1996), 839.  doi: 10.1016/0306-4522(96)00303-X.  Google Scholar

[23]

, I. Wolfram Research,, Mathematica, (2012).   Google Scholar

[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.   Google Scholar

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