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August  2012, 32(8): 2729-2757. doi: 10.3934/dcds.2012.32.2729

Type III excitability, slope sensitivity and coincidence detection

Xiangying Meng 1, , Gemma Huguet 2, and  John Rinzel 3,

1. 

Dynamics and Control, Beihang University, Beijing, China
2. 

Center for Neural Science, New York University, United States
3. 

Center for Neural Science, and Courant Institute of Mathematical Sciences, New York University, United States

Received  May 2011 Revised  July 2011 Published  March 2012

Some neurons in the nervous system do not show repetitive firing for steady currents. For time-varying inputs, they fire once if the input rise is fast enough. This property of phasic firing is known as Type III excitability. Type III excitability has been observed in neurons in the auditory brainstem (MSO), which show strong phase-locking and accurate coincidence detection. In this paper, we consider a Hodgkin-Huxley type model (RM03) that is widely-used for phasic MSO neurons and we compare it with a modification of it, showing tonic behavior. We provide insight into the temporal processing of these neuron models by means of developing and analyzing two reduced models that reproduce qualitatively the properties of the exemplar ones. The geometric and mathematical analysis of the reduced models allows us to detect and quantify relevant features for the temporal computation such as nearness to threshold and a temporal integration window. Our results underscore the importance of Type III excitability for precise coincidence detection.
Keywords: neuronal dynamics., fast-slow systems, phase-locking, coincidence detectors, Type-III excitability.
Mathematics Subject Classification: Primary: 92C20 , Secondary: 37N2.
Citation: Xiangying Meng, Gemma Huguet, John Rinzel. Type III excitability, slope sensitivity and coincidence detection. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2729-2757. doi: 10.3934/dcds.2012.32.2729
References:
[1]

L. R. Bernstein, Auditory processing of interaural timing information: New insights, J. Neurosci. Res., 66, (2001), 1035-1046. doi: 10.1002/jnr.10103.  Google Scholar

[2]

R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, J. Neurophysiol., 94 (2005), 3637-3642. doi: 10.1152/jn.00686.2005.  Google Scholar

[3]

H. M. Brew and I. D. Forsythe, Two voltage-dependent K+ conductances with complementary functions in postsynaptic integration at a central auditory synapse, J. Neurosci., 15 (1995), 8011-8022. Google Scholar

[4]

C. E. Carr and K. M. Macleod, Microseconds matter, PLoS Biol., 8 (2010), e1000405. doi: 10.1371/journal.pbio.1000405.  Google Scholar

[5]

J. R. Clay, D. Paydarfar and D. B. Forger, A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons, J. R. Soc. Interface, 5 (2008), 1421-1428. doi: 10.1098/rsif.2008.0166.  Google Scholar

[6]

D. L. Cook, P. C. Schwindt, L. A. Grande and W. J. Spain, Synaptic depression in the localization of sound, Nature, 421 (2003), 66-70. doi: 10.1038/nature01248.  Google Scholar

[7]

M. L. Day, B. Doiron and J. Rinzel, Subthreshold K+ channel dynamics interact with stimulus spectrum to influence temporal coding in an auditory brain stem model, J. Neurophysiol., 99 (2008), 534-544. doi: 10.1152/jn.00326.2007.  Google Scholar

[8]

R. Dodla, G. Svirskis and J. Rinzel, Well-timed, brief inhibition can promote spiking: Postinhibitory facilitation, J. Neurophysiol., 95 (2006), 2664-2677. doi: 10.1152/jn.00752.2005.  Google Scholar

[9]

R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[10]

R. FitzHugh, Mathematical models of excitation and propagation in nerve, in "Biological Engineering" (ed. H. P. Schwan), McGraw-Hill Book Company, New York, (1969), 1-85. Google Scholar

[11]

Y. Gai, B. Doiron, V. Kotak and J. Rinzel, Noise-gated encoding of slow inputs by auditory brain stem neurons with a low-threshold K+ current, J. Neurophysiol., 102 (2009), 3447-3460. doi: 10.1152/jn.00538.2009.  Google Scholar

[12]

Y. Gai, B. Doiron and J. Rinzel, Slope-based stochastic resonance: How noise enables phasic neurons to encode slow signals, PLoS Comput. Biol., 6 (2010), e1000825, 15 pp.  Google Scholar

[13]

J. M. Goldberg and P. B. Brown, Response of binaural neurons of dog superior olivary complex to dichotic tonal stimuli: Some physiological mechanisms of sound localization, J. Neurophysiol., 32 (1969), 613-636. Google Scholar

[14]

R. Guttman, S. Lewis and J. Rinzel, Control of repetitive firing in squid axon membrane as a model for a neuroneoscillator, J. Physiol. (Lond.), 305 (1980), 377-395. Google Scholar

[15]

A. L. Hodgkin, The local electric changes associated with repetitive action in a non-medullated axon, J. Physiol. (Lond.), 107 (1948), 165-181. Google Scholar

[16]

Eugene M. Izhikevich, "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting," Computational Neuroscience, MIT Press, Cambridge, MA, 2007.  Google Scholar

[17]

B. Lindner and A. Longtin, Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron, J. Theor. Biol., 232 (2005), 505-521. doi: 10.1016/j.jtbi.2004.08.030.  Google Scholar

[18]

Y. H. Liu and X. J. Wang, Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron, J. Comput. Neurosci., 10 (2001), 25-45. doi: 10.1023/A:1008916026143.  Google Scholar

[19]

P. B. Manis and S. O. Marx, Outward currents in isolated ventral cochlear nucleus neurons, J. Neurosci., 11 (1991), 2865-2880. Google Scholar

[20]

X. Meng, Q. Lu and J. Rinzel, Control of firing patterns by two transient potassium currents: Leading spike, latency, bistability, J. Comput. Neurosci., 31 (2010), 117-136. doi: 10.1007/s10827-010-0297-5.  Google Scholar

[21]

X. Y. Meng and J. Rinzel, A two-variable reduction of the Rothman-Manis model for phasic firing, Abstracts of the Thirty-Fourth Annual Mid-Winter Research Meeting of the Association for Research in Otolaryngology, 34 (2011), 154. Google Scholar

[22]

J. Platkiewicz and R. Brette, A threshold equation for action potential initiation, PLoS Comput. Biol., 6 (2010), e1000850, 16 pp.  Google Scholar

[23]

S. A. Prescott and Y. De Koninck, Four cell types with distinctive membrane properties and morphologies in lamina I of the spinal dorsal horn of the adult rat, J. Physiol. (Lond.), 539 (2002), 817-836. doi: 10.1113/jphysiol.2001.013437.  Google Scholar

[24]

S. A. Prescott, Y. De Koninck and T. J. Sejnowski, Biophysical basis for three distinct dynamical mechanisms of action potential initiation, PLoS Comput. Biol., 4 (2008), e1000198, 18 pp.  Google Scholar

[25]

M. Rathouz and L. Trussell, Characterization of outward currents in neurons of the avian nucleus magnocellularis, J. Neurophysiol., 80 (1998), 2824-2835. Google Scholar

[26]

A. D. Reyes, E. W. Rubel and W. J. Spain, In vitro analysis of optimal stimuli for phase-locking and time-delayed modulation of firing in avian nucleus laminaris neurons, J. Neurosci., 16 (1996), 993-1007. Google Scholar

[27]

M. J. Richardson, N. Brunel and V. Hakim, From subthreshold to firing-rate resonance, J. Neurophysiol., 89 (2003), 2538-2554. doi: 10.1152/jn.00955.2002.  Google Scholar

[28]

J. Rinzel, On repetitive activity in nerve, Fed. Proc., 37 (1978), 2793-2802. Google Scholar

[29]

J. Rinzel, Excitation dynamics: Insights from simplified membrane models, Fed. Proc., 44 (1985), 2944-2946. Google Scholar

[30]

J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, in "Methods in Neuronal Modelling: From synapses to Networks" (eds. C. Koch and I. Segev), $2^{nd}$ edition, MIT Press, Cambridge, MA, (1998), 251-291. Google Scholar

[31]

J. Rinzel, D. Terman, X. Wang and B. Ermentrout, Propagating activity patterns in large-scale inhibitory neuronal networks, Science, 279 (1998), 1351-1355. doi: 10.1126/science.279.5355.1351.  Google Scholar

[32]

J. S. Rothman and P. B. Manis, The roles potassium currents play in regulating the electrical activity of ventral cochlear nucleus neurons, J. Neurophysiol., 89 (2003), 3097-3113. doi: 10.1152/jn.00127.2002.  Google Scholar

[33]

J. W. Schnupp and C. E. Carr, On hearing with more than one ear: Lessons from evolution, Nat. Neurosci., 12 (2009), 692-697. doi: 10.1038/nn.2325.  Google Scholar

[34]

L. L. Scott, P. J. Mathews and N. L. Golding, Perisomatic voltage-gated sodium channels actively maintain linear synaptic integration in principal neurons of the medial superior olive, J. Neurosci., 30 (2010), 2039-2050. doi: 10.1523/JNEUROSCI.2385-09.2010.  Google Scholar

[35]

J. P. Segundo and O. Diez Martinez, Dynamic and static hysteresis in crayfish stretch receptors, Biol. Cybern., 52 (1985), 291-296. doi: 10.1007/BF00355750.  Google Scholar

[36]

S. J. Slee, M. H. Higgs, A. L. Fairhall and W. J. Spain, Two-dimensional time coding in the auditory brainstem, J. Neurosci., 25 (2005), 9978-9988. doi: 10.1523/JNEUROSCI.2666-05.2005.  Google Scholar

[37]

G. Svirskis, V. Kotak, D. H. Sanes and J. Rinzel, Enhancement of signal-to-noise ratio and phase locking for small inputs by a low-threshold outward current in auditory neurons, J. Neurosci., 22 (2002), 11019-11025. Google Scholar

[38]

G. Svirskis, V. Kotak, D. H. Sanes and J. Rinzel, Sodium along with low-threshold potassium currents enhance coincidence detection of subthreshold noisy signals in MSO neurons, J. Neurophysiol., 91 (2004), 2465-2473. doi: 10.1152/jn.00717.2003.  Google Scholar

[39]

T. Tateno, A. Harsch and H. P. Robinson, Threshold firing frequency-current relationships of neurons in rat somatosensory cortex: Type 1 and type 2 dynamics, J. Neurophysiol., 92 (2004), 2283-2294. doi: 10.1152/jn.00109.2004.  Google Scholar

[40]

X. J. Wang and G. Buzsaki, Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model, J. Neurosci., 16 (1996), 6402-6413. Google Scholar

show all references

References:
[1]

L. R. Bernstein, Auditory processing of interaural timing information: New insights, J. Neurosci. Res., 66, (2001), 1035-1046. doi: 10.1002/jnr.10103.  Google Scholar

[2]

R. Brette and W. Gerstner, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, J. Neurophysiol., 94 (2005), 3637-3642. doi: 10.1152/jn.00686.2005.  Google Scholar

[3]

H. M. Brew and I. D. Forsythe, Two voltage-dependent K+ conductances with complementary functions in postsynaptic integration at a central auditory synapse, J. Neurosci., 15 (1995), 8011-8022. Google Scholar

[4]

C. E. Carr and K. M. Macleod, Microseconds matter, PLoS Biol., 8 (2010), e1000405. doi: 10.1371/journal.pbio.1000405.  Google Scholar

[5]

J. R. Clay, D. Paydarfar and D. B. Forger, A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons, J. R. Soc. Interface, 5 (2008), 1421-1428. doi: 10.1098/rsif.2008.0166.  Google Scholar

[6]

D. L. Cook, P. C. Schwindt, L. A. Grande and W. J. Spain, Synaptic depression in the localization of sound, Nature, 421 (2003), 66-70. doi: 10.1038/nature01248.  Google Scholar

[7]

M. L. Day, B. Doiron and J. Rinzel, Subthreshold K+ channel dynamics interact with stimulus spectrum to influence temporal coding in an auditory brain stem model, J. Neurophysiol., 99 (2008), 534-544. doi: 10.1152/jn.00326.2007.  Google Scholar

[8]

R. Dodla, G. Svirskis and J. Rinzel, Well-timed, brief inhibition can promote spiking: Postinhibitory facilitation, J. Neurophysiol., 95 (2006), 2664-2677. doi: 10.1152/jn.00752.2005.  Google Scholar

[9]

R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[10]

R. FitzHugh, Mathematical models of excitation and propagation in nerve, in "Biological Engineering" (ed. H. P. Schwan), McGraw-Hill Book Company, New York, (1969), 1-85. Google Scholar

[11]

Y. Gai, B. Doiron, V. Kotak and J. Rinzel, Noise-gated encoding of slow inputs by auditory brain stem neurons with a low-threshold K+ current, J. Neurophysiol., 102 (2009), 3447-3460. doi: 10.1152/jn.00538.2009.  Google Scholar

[12]

Y. Gai, B. Doiron and J. Rinzel, Slope-based stochastic resonance: How noise enables phasic neurons to encode slow signals, PLoS Comput. Biol., 6 (2010), e1000825, 15 pp.  Google Scholar

[13]

J. M. Goldberg and P. B. Brown, Response of binaural neurons of dog superior olivary complex to dichotic tonal stimuli: Some physiological mechanisms of sound localization, J. Neurophysiol., 32 (1969), 613-636. Google Scholar

[14]

R. Guttman, S. Lewis and J. Rinzel, Control of repetitive firing in squid axon membrane as a model for a neuroneoscillator, J. Physiol. (Lond.), 305 (1980), 377-395. Google Scholar

[15]

A. L. Hodgkin, The local electric changes associated with repetitive action in a non-medullated axon, J. Physiol. (Lond.), 107 (1948), 165-181. Google Scholar

[16]

Eugene M. Izhikevich, "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting," Computational Neuroscience, MIT Press, Cambridge, MA, 2007.  Google Scholar

[17]

B. Lindner and A. Longtin, Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron, J. Theor. Biol., 232 (2005), 505-521. doi: 10.1016/j.jtbi.2004.08.030.  Google Scholar

[18]

Y. H. Liu and X. J. Wang, Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron, J. Comput. Neurosci., 10 (2001), 25-45. doi: 10.1023/A:1008916026143.  Google Scholar

[19]

P. B. Manis and S. O. Marx, Outward currents in isolated ventral cochlear nucleus neurons, J. Neurosci., 11 (1991), 2865-2880. Google Scholar

[20]

X. Meng, Q. Lu and J. Rinzel, Control of firing patterns by two transient potassium currents: Leading spike, latency, bistability, J. Comput. Neurosci., 31 (2010), 117-136. doi: 10.1007/s10827-010-0297-5.  Google Scholar

[21]

X. Y. Meng and J. Rinzel, A two-variable reduction of the Rothman-Manis model for phasic firing, Abstracts of the Thirty-Fourth Annual Mid-Winter Research Meeting of the Association for Research in Otolaryngology, 34 (2011), 154. Google Scholar

[22]

J. Platkiewicz and R. Brette, A threshold equation for action potential initiation, PLoS Comput. Biol., 6 (2010), e1000850, 16 pp.  Google Scholar

[23]

S. A. Prescott and Y. De Koninck, Four cell types with distinctive membrane properties and morphologies in lamina I of the spinal dorsal horn of the adult rat, J. Physiol. (Lond.), 539 (2002), 817-836. doi: 10.1113/jphysiol.2001.013437.  Google Scholar

[24]

S. A. Prescott, Y. De Koninck and T. J. Sejnowski, Biophysical basis for three distinct dynamical mechanisms of action potential initiation, PLoS Comput. Biol., 4 (2008), e1000198, 18 pp.  Google Scholar

[25]

M. Rathouz and L. Trussell, Characterization of outward currents in neurons of the avian nucleus magnocellularis, J. Neurophysiol., 80 (1998), 2824-2835. Google Scholar

[26]

A. D. Reyes, E. W. Rubel and W. J. Spain, In vitro analysis of optimal stimuli for phase-locking and time-delayed modulation of firing in avian nucleus laminaris neurons, J. Neurosci., 16 (1996), 993-1007. Google Scholar

[27]

M. J. Richardson, N. Brunel and V. Hakim, From subthreshold to firing-rate resonance, J. Neurophysiol., 89 (2003), 2538-2554. doi: 10.1152/jn.00955.2002.  Google Scholar

[28]

J. Rinzel, On repetitive activity in nerve, Fed. Proc., 37 (1978), 2793-2802. Google Scholar

[29]

J. Rinzel, Excitation dynamics: Insights from simplified membrane models, Fed. Proc., 44 (1985), 2944-2946. Google Scholar

[30]

J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, in "Methods in Neuronal Modelling: From synapses to Networks" (eds. C. Koch and I. Segev), $2^{nd}$ edition, MIT Press, Cambridge, MA, (1998), 251-291. Google Scholar

[31]

J. Rinzel, D. Terman, X. Wang and B. Ermentrout, Propagating activity patterns in large-scale inhibitory neuronal networks, Science, 279 (1998), 1351-1355. doi: 10.1126/science.279.5355.1351.  Google Scholar

[32]

J. S. Rothman and P. B. Manis, The roles potassium currents play in regulating the electrical activity of ventral cochlear nucleus neurons, J. Neurophysiol., 89 (2003), 3097-3113. doi: 10.1152/jn.00127.2002.  Google Scholar

[33]

J. W. Schnupp and C. E. Carr, On hearing with more than one ear: Lessons from evolution, Nat. Neurosci., 12 (2009), 692-697. doi: 10.1038/nn.2325.  Google Scholar

[34]

L. L. Scott, P. J. Mathews and N. L. Golding, Perisomatic voltage-gated sodium channels actively maintain linear synaptic integration in principal neurons of the medial superior olive, J. Neurosci., 30 (2010), 2039-2050. doi: 10.1523/JNEUROSCI.2385-09.2010.  Google Scholar

[35]

J. P. Segundo and O. Diez Martinez, Dynamic and static hysteresis in crayfish stretch receptors, Biol. Cybern., 52 (1985), 291-296. doi: 10.1007/BF00355750.  Google Scholar

[36]

S. J. Slee, M. H. Higgs, A. L. Fairhall and W. J. Spain, Two-dimensional time coding in the auditory brainstem, J. Neurosci., 25 (2005), 9978-9988. doi: 10.1523/JNEUROSCI.2666-05.2005.  Google Scholar

[37]

G. Svirskis, V. Kotak, D. H. Sanes and J. Rinzel, Enhancement of signal-to-noise ratio and phase locking for small inputs by a low-threshold outward current in auditory neurons, J. Neurosci., 22 (2002), 11019-11025. Google Scholar

[38]

G. Svirskis, V. Kotak, D. H. Sanes and J. Rinzel, Sodium along with low-threshold potassium currents enhance coincidence detection of subthreshold noisy signals in MSO neurons, J. Neurophysiol., 91 (2004), 2465-2473. doi: 10.1152/jn.00717.2003.  Google Scholar

[39]

T. Tateno, A. Harsch and H. P. Robinson, Threshold firing frequency-current relationships of neurons in rat somatosensory cortex: Type 1 and type 2 dynamics, J. Neurophysiol., 92 (2004), 2283-2294. doi: 10.1152/jn.00109.2004.  Google Scholar

[40]

X. J. Wang and G. Buzsaki, Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model, J. Neurosci., 16 (1996), 6402-6413. Google Scholar

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