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Dynamics of a delay differential equation with multiple state-dependent delays
Type III excitability, slope sensitivity and coincidence detection
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 |
References:
[1] |
L. R. Bernstein, Auditory processing of interaural timing information: New insights,, J. Neurosci. Res., 66 (2001), 1035.
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.
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. Google Scholar |
[4] |
C. E. Carr and K. M. Macleod, Microseconds matter,, PLoS Biol., 8 (2010).
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.
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.
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.
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.
doi: 10.1152/jn.00752.2005. |
[9] |
R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445.
doi: 10.1016/S0006-3495(61)86902-6. |
[10] |
R. FitzHugh, Mathematical models of excitation and propagation in nerve,, in, (1969), 1. 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.
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).
|
[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. 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. 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. Google Scholar |
[16] |
Eugene M. Izhikevich, "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting,", Computational Neuroscience, (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.
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.
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. 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.
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). Google Scholar |
[22] |
J. Platkiewicz and R. Brette, A threshold equation for action potential initiation,, PLoS Comput. Biol., 6 (2010).
|
[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.
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).
|
[25] |
M. Rathouz and L. Trussell, Characterization of outward currents in neurons of the avian nucleus magnocellularis,, J. Neurophysiol., 80 (1998), 2824. 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. Google Scholar |
[27] |
M. J. Richardson, N. Brunel and V. Hakim, From subthreshold to firing-rate resonance,, J. Neurophysiol., 89 (2003), 2538.
doi: 10.1152/jn.00955.2002. |
[28] |
J. Rinzel, On repetitive activity in nerve,, Fed. Proc., 37 (1978), 2793. Google Scholar |
[29] |
J. Rinzel, Excitation dynamics: Insights from simplified membrane models,, Fed. Proc., 44 (1985), 2944. Google Scholar |
[30] |
J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations,, in, (1998), 251. 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.
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.
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.
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.
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.
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.
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. 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.
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.
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. Google Scholar |
show all references
References:
[1] |
L. R. Bernstein, Auditory processing of interaural timing information: New insights,, J. Neurosci. Res., 66 (2001), 1035.
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.
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. Google Scholar |
[4] |
C. E. Carr and K. M. Macleod, Microseconds matter,, PLoS Biol., 8 (2010).
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.
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.
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.
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.
doi: 10.1152/jn.00752.2005. |
[9] |
R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445.
doi: 10.1016/S0006-3495(61)86902-6. |
[10] |
R. FitzHugh, Mathematical models of excitation and propagation in nerve,, in, (1969), 1. 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.
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).
|
[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. 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. 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. Google Scholar |
[16] |
Eugene M. Izhikevich, "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting,", Computational Neuroscience, (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.
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.
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. 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.
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). Google Scholar |
[22] |
J. Platkiewicz and R. Brette, A threshold equation for action potential initiation,, PLoS Comput. Biol., 6 (2010).
|
[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.
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).
|
[25] |
M. Rathouz and L. Trussell, Characterization of outward currents in neurons of the avian nucleus magnocellularis,, J. Neurophysiol., 80 (1998), 2824. 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. Google Scholar |
[27] |
M. J. Richardson, N. Brunel and V. Hakim, From subthreshold to firing-rate resonance,, J. Neurophysiol., 89 (2003), 2538.
doi: 10.1152/jn.00955.2002. |
[28] |
J. Rinzel, On repetitive activity in nerve,, Fed. Proc., 37 (1978), 2793. Google Scholar |
[29] |
J. Rinzel, Excitation dynamics: Insights from simplified membrane models,, Fed. Proc., 44 (1985), 2944. Google Scholar |
[30] |
J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations,, in, (1998), 251. 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.
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.
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.
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.
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.
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.
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. 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.
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.
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. Google Scholar |
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