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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 and Continuous Dynamical Systems, 2012, 32 (8) : 2729-2757. doi: 10.3934/dcds.2012.32.2729
References:
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L. R. Bernstein, Auditory processing of interaural timing information: New insights, J. Neurosci. Res., 66, (2001), 1035-1046. doi: 10.1002/jnr.10103.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[40]

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

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