There is growing evidence that the dynamics of biological systems that appear to be exponential over short time courses are in some cases better described over the long-term by power-law dynamics. A model of rate adaptation at the synapse between inner hair cells and auditory-nerve (AN) fibers that includes both exponential and power-law dynamics is presented here. Exponentially adapting components with rapid and short-term time constants, which are mainly responsible for shaping onset responses, are followed by two parallel paths with power-law adaptation that provide slowly and rapidly adapting responses. The slowly adapting power-law component significantly improves predictions of the recovery of the AN response after stimulus offset. The faster power-law adaptation is necessary to account for the “additivity” of rate in response to stimuli with amplitude increments. The proposed model is capable of accurately predicting several sets of AN data, including amplitude-modulation transfer functions, long-term adaptation, forward masking, and adaptation to increments and decrements in the amplitude of an ongoing stimulus.
REFERENCES
1.
Abbas
, P. J.
(1979
). “Effects of stimulus frequency on adaptation in auditory-nerve fibers
,” J. Acoust. Soc. Am.
65
, 162
–165
.2.
Beurg
, M.
, Nam
, J.-H.
, Crawford
, A.
, and Fettiplace
, R.
(2008
). “The actions of calcium on hair bundle mechanics in mammalian cochlear hair cells
,” Biophys. J.
94
, 2639
–2653
.3.
Breebart
, J.
, van de Par
, S.
, and Kohlrausch
, A.
(2001
). “Binaural processing model based on contralateral inhibition. III. Dependence on temporal parameters
,” J. Acoust. Soc. Am.
110
, 1105
–1117
.4.
Brown
, M. C.
, and Stein
, R. B.
(1966
). “Quantitative studies on the slowly adapting stretch receptor of the crayfish
,” Kybernetik
3
, 175
–185
.5.
Bruce
, I. C.
, Sachs
, M. B.
, and Young
, E. D.
(2003
). “An auditory-periphery model of the effects of acoustic trauma on auditory nerve responses
,” J. Acoust. Soc. Am.
113
, 369
–388
.6.
Camera
, G. L.
, Rauch
, A.
, Thurbon
, D.
, Luscher
, H. R.
, Senn
, W.
, and Fusi
, S.
(2006
). “Multiple time scales of temporal response in pyramidal and fast spiking cortical neurons
,” J. Neurophysiol.
96
, 3448
–3464
.7.
Carney
, L. H.
(1993
). “A model for the responses of low-frequency auditory-nerve fibers in cat
,” J. Acoust. Soc. Am.
93
, 401
–417
.8.
Chapman
, K. M.
, and Smith
, R. S.
(1963
). “A linear transfer function underlying impulse frequency modulation in a cockroach mechanoreceptor
,” Nature (London)
197
, 699
–700
.9.
Cooke
, M. P.
(1986
). “A computer model of peripheral auditory processing
,” Speech Commun.
5
, 261
–281
.10.
Dau
, T.
, Kollmeier
, B.
, and Kohlrausch
, A.
(1997
). “Modeling auditory processing of amplitude modulation. I. Detection and masking with narrow-band carriers
,” J. Acoust. Soc. Am.
102
, 2892
–2905
.11.
Dau
, T.
, Püschel
, D.
, and Kohlrausch
, A.
(1996a
). “A quantitative model of the effective signal processing in the auditory system. I. Model structure
,” J. Acoust. Soc. Am.
99
, 3615
–3622
.12.
Dau
, T.
, Püschel
, D.
, and Kohlrausch
, A.
(1996b
). “A quantitative model of the effective signal processing in the auditory system. II. Simulations and measurements
,” J. Acoust. Soc. Am.
99
, 3623
–3631
.13.
Dean
, I.
, Harper
, N. S.
, and McAlpine
, D.
(2005
). “Neural population coding of sound level adapts to stimulus statistics
,” Nat. Neurosci.
8
, 1684
–1689
.14.
Delgutte
, B.
(1980
). “Representation of speech-like sounds in the discharge patterns of auditory-nerve fibers
,” J. Acoust. Soc. Am.
68
, 843
–857
.15.
Drew
, P. J.
, and Abbott
, L. F.
(2006
). “Models and properties of power-law adaptation in neural system
,” J. Neurophysiol.
96
, 826
–833
.16.
Ewert
, S. D.
, Verhey
, J. L.
, and Dau
, T.
(2002
). “Spectro-temporal processing in the envelope-frequency domain
,” J. Acoust. Soc. Am.
112
, 2921
–2931
.17.
Fairhall
, A. L.
, Lewen
, G. D.
, Bialek
, W.
, and de Ruyter van Steveninck
, R. R.
(2001
). “Efficiency and ambiguity in an adaptive neural code
,” Nature (London)
412
, 787
–792
.18.
French
, A. S.
(1984
). “The receptor potential and adaptation in the cockroach tactile spine
,” J. Neurosci.
4
, 2063
–2068
.19.
French
, A. S.
, and Torkkeli
, P. H.
(2008
). “The power law of sensory adaptation: Simulation by a model of excitability in spider mechanoreceptor neurons
,” Ann. Biomed. Eng.
36
, 153
–161
.20.
Furukawa
, T.
, and Matsuura
, S.
(1978
). “Adaptive rundown of excitatory postsynaptic potentials at synapses between hair cells and eight nerve fibers in the goldfish
,” J. Physiol. (London)
276
, 193
–209
.21.
Furukawa
, T.
, Mikuki
, K.
, and Matsuura
, S.
(1982
). “Quantal analysis of a decremental response at hair cell-afferent fiber synapse in the goldfish sacculus
,” J. Physiol. (London)
322
, 181
–195
.22.
Goutman
, J. D.
, and Glowatzki
, E.
(2007
). “Time course and calcium dependence of transmitter release at a single ribbon synapse
,” Proc. Natl. Acad. Sci. U.S.A.
104
, 16341
–16346
.23.
Greenwood
, D. D.
, and Joris
, P. X.
(1996
). “Mechanical and temporal filtering as codeterminants of the response by cat primary fibers to the amplitude-modulated signal
,” J. Acoust. Soc. Am.
99
, 1029
–1039
.24.
Hanna
, T. E.
, Robinson
, D. E.
, Shiffrin
, R. M.
, and Gilkey
, R. H.
(1982
). “Forward masking of diotic and dichotic clicks by noise
,” J. Acoust. Soc. Am.
72
, 1171
–1177
.25.
Harris
, D. M.
, and Dallos
, P.
(1979
). “Forward masking of auditory nerve fiber responses
,” J. Neurophysiol.
42
, 1083
–1107
.26.
Hewitt
, M. J.
, and Meddis
, R.
(1991
). “An evaluation of eight computer models of mammalian inner hair-cell function
,” J. Acoust. Soc. Am.
90
, 904
–917
.27.
Jackson
, B. S.
(2003
). “Consequences of long-range temporal dependence in neural spike trains for theories of coding and processing
,” Ph.D. thesis, Syracuse University
, Syracuse, NY (2003
).28.
Jackson
, B. S.
, and Carney
, L. H.
(2005
). “The spontaneous-rate histogram of the auditory nerve can be explained by only two or three spontaneous rates and long-range dependence
,” J. Assoc. Res. Otolaryngol.
6
, 148
–159
.29.
Jia
, S.
, Dallos
, P.
, and He
, D. Z.
(2007
). “Mechanoelectric transduction of adult inner hair cells
,” J. Neurosci.
27
, 1006
–1014
.30.
Johnson
, D.
(1980
). “The relationship between spike rate and synchrony in responses to auditory-nerve fibers to single tones
,” J. Acoust. Soc. Am.
68
, 1115
–1122
.31.
Joris
, P. X.
(2003
). “Interaural time sensitivity dominated by cochlea-induced envelope patterns
,” J. Neurosci.
23
, 6345
–6350
.32.
Joris
, P. X.
, Carney
, L. H.
, Smith
, P. H.
, and Yin
, T. C. T.
(1994
). “Enhancement of neural synchronization in the anteroventral cochlear nucleus I: Responses to tones at the characteristic frequency
,” J. Neurophysiol.
71
, 1022
–1036
.33.
Joris
, P. X.
, and Yin
, T. C. T.
(1992
). “Responses to amplitude-modulated tones in the auditory nerve of the cat
,” J. Acoust. Soc. Am.
91
, 215
–232
.34.
Kelly
, O. E.
, Johnson
, D. H.
, Delgutte
, B.
, and Cariani
, P.
(1996
). “Fractal noise strength in auditory-nerve fiber recordings
,” J. Acoust. Soc. Am.
99
, 2210
–2220
.35.
Kiang
, N. Y.-S.
(1965
). “Discharge patterns of single fibers in the cat’s auditory nerve
,” M.I.T. Research Monograph No. 35
(Technology Press
, Boston, MA).36.
Kiang
, N. Y.-S.
(1990
). “Curious oddments of auditory-nerve studies
,” Hear. Res.
49
, 1
–16
.37.
Kros
, C. J.
, and Crawford
, A. C.
(1990
). “Potassium currents in inner hair cells isolated from the guinea-pig cochlea
,” J. Physiol. (London)
421
, 263
–291
.38.
Leopold
, D. A.
, Murayama
, Y.
, and Logothetis
, N.
(2003
). “Very slow activity fluctuations in monkey visual cortex: Implications for functional brain imaging
,” Cereb. Cortex
13
, 422
–433
.39.
Liberman
, M. C.
(1978
). “Auditory-nerve response from cats raised in a low-noise chamber
,” J. Acoust. Soc. Am.
63
, 442
–455
.40.
Louage
, D. H. G.
, Heijden
, M. v. d.
, and Joris
, P. X.
(2004
). “Temporal properties of responses to broadband noise in the auditory nerve
,” J. Neurophysiol.
91
, 2051
–2065
.41.
Lundstrom
, B. N.
, Higgs
, M. H.
, Spain
, W. J.
, and Fairhall
, A. L.
(2008
). “Fractional differentiation by neocortical pyramidal neurons
,” Nat. Neurosci.
11
, 1335
–1342
.42.
Meddis
, R.
(1986
). “Simulation of mechanical to neural transduction in the auditory receptor
,” J. Acoust. Soc. Am.
79
, 702
–711
.43.
Meddis
, R.
(1988
). “Simulation of auditory-neural transduction: Further studies
,” J. Acoust. Soc. Am.
83
, 1056
–1063
.44.
Meddis
, R.
, and O’Mard
, L. P.
(2005
). “A computer model of the auditory nerve response to forward masking stimuli
,” J. Acoust. Soc. Am.
117
, 3787
–3798
.45.
Miller
, R. L.
, Schilling
, J. R.
, Franck
, K. R.
, and Young
, E. D.
(1997
). “Effects of acoustic trauma on the representation of the vowel /ε/ in cat auditory nerve fibers
,” J. Acoust. Soc. Am.
101
, 3602
–3616
.46.
Moser
, T.
, and Beutner
, D.
(2000
). “Kinetics of exocytosis and endocytosis at the cochlear inner hair cell afferent synapse of the mouse
,” Proc. Natl. Acad. Sci. U.S.A.
97
, 883
–888
.47.
Müller
, M.
, and Robertson
, D.
(1991
). “Relationship between tone burst discharge pattern and spontaneous firing rate of auditory nerve fibers in the guinea-pig
,” Hear. Res.
57
, 63
–70
.48.
Nelson
, P. C.
, and Carney
, L. H.
(2004
). “A phenomenological model of peripheral and central neural responses to amplitude-modulated tones
,” J. Acoust. Soc. Am.
116
, 2173
–2186
.49.
Nelson
, P. C.
, Smith
, Z. M.
, and Young
, E. D.
(2009
). “Wide dynamic range forward suppression in marmoset inferior colliculus neurons is generated centrally and accounts for perceptual masking
,” J. Neurosci.
29
, 2553
–2562
.50.
Oono
, Y.
, and Sujaku
, Y.
(1974
). “A probabilistic model for discharge patterns of auditory nerve fibers
,” Trans. Inst. Elect. Comm. Eng. (Japan)
57
, 35
–36
.51.
Oono
, Y.
, and Sujaku
, Y.
(1975
). “A model for automatic gain control observed in the firing of primary auditory neurons
,” Trans. Inst. Elect. Comm. Eng.
58
, 61
–62
.52.
Payton
, K. L.
(1988
). “Vowel processing by a model of the auditory periphery: A comparison to eighth-nerve responses
,” J. Acoust. Soc. Am.
83
, 145
–162
.53.
Raman
, I. M.
, Zhang
, S.
, and Trussell
, L. O.
(1994
). “Pathway-specific variants of AMPA receptors and their contribution to neuronal signaling
,” J. Neurosci.
14
(18
), 4998
–5010
.54.
Relkin
, E. M.
, and Doucet
, J. R.
(1991
). “Recovery from prior stimulation. I: Relationship to spontaneous firing rates of primary auditory neurons
,” Hear. Res.
55
, 215
–222
.55.
Relkin
, E. M.
, and Turner
, C. W.
(1988
). “A reexamination of forward masking in the auditory nerve
,” J. Acoust. Soc. Am.
84
, 584
–591
.56.
Rhode
, W. S.
, and Smith
, P. H.
(1985
). “Characteristics of tone-pip response patterns in relationship to spontaneous rate in cat auditory nerve fibers
,” Hear. Res.
18
, 159
–168
.57.
Ross
, S.
(1982
). “A model of the hair cell-primary fiber complex
,” J. Acoust. Soc. Am.
71
, 926
–941
.58.
Ross
, S.
(1996
). “A functional model of the hair cell-primary fiber complex
,” J. Acoust. Soc. Am.
99
, 2221
–2238
.59.
Schnee
, M. E.
, Lawton
, D. M.
, Furness
, D. N.
, Benke
, T. A.
, and Ricci
, A. J.
(2005
). “Auditory hair cell-afferent fiber synapses are specialized to operate at their best frequencies
,” Neuron
47
, 243
–254
.60.
Schroeder
, M. R.
, and Hall
, J. L.
(1974
). “Model for mechanical to neural transduction in the auditory receptor
,” J. Acoust. Soc. Am.
55
, 1055
–1060
.61.
Schwid
, H. A.
, and Geisler
, C. D.
(1982
). “Multiple reservoir model of neurotransmitter release by a cochlear inner hair cell
,” J. Acoust. Soc. Am.
72
, 1435
–1440
.62.
Shannon
, R. V.
, Zeng
, F.-G.
, Wygonski
, J.
, Kamath
, V.
, and Ekelid
, M.
(1995
). “Speech recognition with primarily temporal cues
,” Science
270
, 303
–304
.63.
Siebert
, W. M.
, and Gambardella
, G.
(1968
). “Phenomenological model for a form of adaptation in primary auditory-nerve fibers
,” RLE QPR Communications Biophysics
88
, 330
–334
.64.
Smirnakis
, S. M.
, Berry
, M. J.
, Warland
, D. K.
, Bialek
, W.
, and Meister
, M.
(1997
). “Adaptation of retinal processing to image contrast and spatial scale
,” Nature (London)
386
, 69
–73
.65.
Smith
, R. L.
(1977
). “Short-term adaptation in single auditory-nerve fibers: Some post-stimulatory effects
,” J. Neurophysiol.
40
, 1098
–1112
.66.
Smith
, R. L.
(1988
). “Encoding of sound intensity by auditory neurons
,” in Auditory Function: Neurobiological Bases of Hearing
, edited by G. M.
Edelman
, W. E.
Gall
, and W. M.
Cowan
(Wiley
, New York
), pp. 243
–274
.67.
Smith
, R. L.
, and Brachman
, M. L.
(1982
). “Adaptation in auditory-nerve fibers: A revised model
,” Biol. Cybern.
44
, 107
–120
.68.
Smith
, R. L.
, Brachman
, M. L.
, and Frisina
, R. D.
(1985
). “Sensitivity of auditory-nerve fibers to changes in intensity: A dichotomy between decrements and increments
,” J. Acoust. Soc. Am.
78
, 1310
–1316
.69.
Smith
, R. L.
, and Zwislocki
, J. J.
(1975
). “Short-term adaptation and incremental responses of single auditory-nerve fibers
,” Biol. Cybern.
17
, 169
–182
.70.
Sumner
, C. J.
, Lopez-Poveda
, E. A.
, O’Mard
, L. P.
, and Meddis
, R.
(2002
). “A revised model of the inner-hair cell and auditory-nerve complex
,” J. Acoust. Soc. Am.
111
, 2178
–2188
.71.
Sumner
, C. J.
, Lopez-Poveda
, E. A.
, O’Mard
, L. P.
, and Meddis
, R.
(2003
). “Adaptation in a revised inner-hair cell model
,” J. Acoust. Soc. Am.
113
, 893
–901
.73.
Teich
, M. C.
(1989
). “Fractal character of the auditory neural spike train
,” IEEE Trans. Biomed. Eng.
36
, 150
–160
.74.
Teich
, M. C.
, and Lowen
, S. B.
(1994
). “Fractal patterns in auditory nerve-spike trains
,” IEEE Eng. Med. Biol. Mag.
13
, 197
–202
.75.
Thorson
, J.
, and Biederman-Thorson
, M.
(1974
). “Distributed relaxation processes in sensory adaptation
,” Science
183
, 161
–172
.76.
Toib
, A.
, Lyakhov
, V.
, and Marom
, S.
(1998
). “Interaction between duration of activity and time course of recovery from slow inactivation in mammalian brain channels
,” J. Neurosci.
18
, 1893
–1903
.77.
Ulanovsky
, N.
, Las
, L.
, Farkas
, D.
, and Nelken
, I.
(2004
). “Multiple timescales of adaptation in auditory cortex neurons
,” J. Neurosci.
24
, 10440
–10453
.78.
Watkins
, P. V.
, and Barbour
, D. L.
(2008
). “Specialized neuronal adaptation for preserving input sensitivity
,” Norelco Rep.
11
, 1259
–1261
.79.
Westerman
, L. A.
(1985
). “Adaptation and recovery of auditory nerve responses
,” Special Report No. ISR-S-24, Syracuse University
, Syracuse, NY.80.
Westerman
, L. A.
, and Smith
, R. L.
(1987
). “Conservation of adapting components in auditory-nerve responses
,” J. Acoust. Soc. Am.
81
, 680
–691
.81.
Westerman
, L. A.
, and Smith
, R. L.
(1988
). “A diffusion model of the transient response of the cochlear inner hair cell synapse
,” J. Acoust. Soc. Am.
83
, 2266
–2276
.82.
Wixted
, J. T.
, and Ebbesen
, E.
(1997
). “Genuine power curves in forgetting
,” Mem. Cognit.
25
, 731
–739
.83.
Xu
, Z.
, Payne
, J. R.
, and Nelson
, M. E.
(1996
). “Logarithmic time course of sensory adaptation in electrosensory afferent nerve fibers in a weakly electric fish
,” J. Neurophysiol.
76
, 2020
–2032
.84.
Yates
, G. K.
(1987
). “Dynamic effects in the input/output relationship of auditory nerve
,” Hear. Res.
27
, 221
–230
.85.
Yates
, G. K.
, Robertson
, D.
, and Johnstone
, B. M.
(1985
). “Very rapid adaptation in the guinea pig auditory nerve
,” Hear. Res.
17
, 1
–12
.86.
Young
, E. D.
, and Sachs
, M. B.
(1973
). “Recovery from sound exposure in auditory-nerve fibers
,” J. Acoust. Soc. Am.
54
, 1535
–1543
.87.
Zeddies
, D. G.
, and Siegel
, J. H.
(2004
). “A biophysical model of an inner hair cell
,” J. Acoust. Soc. Am.
116
, 426
–441
.88.
Zhang
, X.
, and Carney
, L. H.
(2005
). “Analysis of models for the synapse between the inner hair cell and the auditory nerve
,” J. Acoust. Soc. Am.
118
, 1540
–1553
.89.
Zhang
, X.
, Heinz
, M. G.
, Bruce
, I. C.
, and Carney
, L. H.
(2001
). “A phenomenological model for the responses of auditory-nerve fibers. I. Nonlinear tuning with compression and suppression
,” J. Acoust. Soc. Am.
109
, 648
–670
.90.
Zhang
, F.
, Miller
, C. A.
, Robinson
, B. K.
, Abbas
, P. J.
, and Hu
, N.
(2007
). “Changes across time in spike rate and spike amplitude of auditory nerve fibers simulated by electric pulse trains
,” J. Assoc. Res. Otolaryngol.
8
, 356
–372
.91.
Zilany
, M. S. A.
, and Bruce
, I. C.
(2006
). “Modeling auditory-nerve responses for high sound pressure levels in the normal and impaired auditory periphery
,” J. Acoust. Soc. Am.
120
, 1446
–1466
.92.
Zilany
, M. S. A.
, and Bruce
, I. C.
(2007
). “Representation of the vowel/in normal and impaired auditory nerve fibers: Model predictions of responses in cats
,” J. Acoust. Soc. Am.
122
, 402
–417
.93.
Zwicker
, E.
(1984
). “Dependence of post-masking on masker duration and its relation to temporal effects in loudness
,” J. Acoust. Soc. Am.
75
, 219
–223
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