-
Previous Article
Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold
- DCDS Home
- This Issue
-
Next Article
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), 1035-1046.
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. |
[1] |
Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123 |
[2] |
Stefan Martignoli, Ruedi Stoop. Phase-locking and Arnold coding in prototypical network topologies. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 145-162. doi: 10.3934/dcdsb.2008.9.145 |
[3] |
Timothy J. Lewis. Phase-locking in electrically coupled non-leaky integrate-and-fire neurons. Conference Publications, 2003, 2003 (Special) : 554-562. doi: 10.3934/proc.2003.2003.554 |
[4] |
Xiaoxue Zhao, Zhuchun Li, Xiaoping Xue. Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring. Networks and Heterogeneous Media, 2018, 13 (2) : 323-337. doi: 10.3934/nhm.2018014 |
[5] |
Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112 |
[6] |
Ryotaro Tsuneki, Shinji Doi, Junko Inoue. Generation of slow phase-locked oscillation and variability of the interspike intervals in globally coupled neuronal oscillators. Mathematical Biosciences & Engineering, 2014, 11 (1) : 125-138. doi: 10.3934/mbe.2014.11.125 |
[7] |
Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257 |
[8] |
Ramon Quintanilla, Reinhard Racke. Stability in thermoelasticity of type III. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 383-400. doi: 10.3934/dcdsb.2003.3.383 |
[9] |
Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure and Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219 |
[10] |
Chunhua Shan. Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1447-1469. doi: 10.3934/dcdsb.2021097 |
[11] |
Ilya Schurov. Duck farming on the two-torus: Multiple canard cycles in generic slow-fast systems. Conference Publications, 2011, 2011 (Special) : 1289-1298. doi: 10.3934/proc.2011.2011.1289 |
[12] |
Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621 |
[13] |
C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603 |
[14] |
Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic and Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025 |
[15] |
Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350 |
[16] |
Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433 |
[17] |
Shigeki Akiyama. Strong coincidence and overlap coincidence. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5223-5230. doi: 10.3934/dcds.2016027 |
[18] |
Jie Xu, Yu Miao, Jicheng Liu. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2233-2256. doi: 10.3934/dcdsb.2015.20.2233 |
[19] |
Seung-Yeal Ha, Dohyun Kim, Jinyeong Park. Fast and slow velocity alignments in a Cucker-Smale ensemble with adaptive couplings. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4621-4654. doi: 10.3934/cpaa.2020209 |
[20] |
Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]