Abstract
The ability to estimate and produce appropriately timed responses is central to many behaviors including speaking, dancing, and playing a musical instrument. A classical framework for estimating or producing a time interval is the pacemaker-accumulator model in which pulses of a pacemaker are counted and compared to a stored representation. However, the neural mechanisms for how these pulses are counted remain an open question. The presence of noise and stochasticity further complicates the picture. We present a biophysical model of how to keep count of a pacemaker in the presence of various forms of stochasticity using a system of bistable Wilson-Cowan units asymmetrically connected in a one-dimensional array; all units receive the same input pulses from a central clock but only one unit is active at any point in time. With each pulse from the clock, the position of the activated unit changes thereby encoding the total number of pulses emitted by the clock. This neural architecture maps the counting problem into the spatial domain, which in turn translates count to a time estimate. We further extend the model to a hierarchical structure to be able to robustly achieve higher counts.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmadian Y, Rubin DB, Miller KD (2013) Analysis of the stabilized supralinear network. Neural Comput 25:1994–2037. https://doi.org/10.1162/NECO
Almeida R, Ledberg A (2010) A biologically plausible model of time-scale invariant interval timing. J Comput Neurosci 28:155–175. https://doi.org/10.1007/s10827-009-0197-8
Amit DJ (1988) Neural Networks Counting Chimes Author ( s ): Daniel J . Amit Source : Proceedings of the National Academy of Sciences of the United States of Published by : National Academy of Sciences Stable URL : https://www.jstor.org/stable/31085 REFERENCES Linked re. 85:2141–2145
Aschersleben G (2002) Temporal control of movements in sensorimotor synchronization. Brain Cogn 48:66–79. https://doi.org/10.1006/brcg.2001.1304
Bengtsson SL, Ullén F, Henrik Ehrsson H et al (2009) Listening to rhythms activates motor and premotor cortices. Cortex 45:62–71. https://doi.org/10.1016/j.cortex.2008.07.002
Bennett MR, Gibson WG, Robinson J (1997) Probabilistic secretion of quanta and the synaptosecretosome hypothesis: evoked release at active zones of varicosities, boutons, and endplates. Biophys J 73:1815–1829. https://doi.org/10.1016/S0006-3495(97)78212-2
Borisyuk RM, Kirillov AB (1992) Bifurcation analysis of a neural network model. Biol Cybern 66:319–325. https://doi.org/10.1007/BF00203668
Bose A, Byrne Á, Rinzel J (2019) A neuromechanistic model for rhythmic beat generation. PLOS Comput Biol 15:e1006450. https://doi.org/10.1371/journal.pcbi.1006450
Brody CD, Hernández A, Zainos A, Romo R (2003) Timing and neural encoding of somatosensory parametric working memory in macaque prefrontal cortex. Cereb Cortex 13:1196–1207. https://doi.org/10.1093/CERCOR/BHG100
Buhusi CV, Meck WH (2005) What makes us tick? Functional and neural mechanisms of interval timing. Nat Rev Neurosci 6:755–765
Cao R, Braun J, Mattia M (2014) Stochastic accumulation by cortical columns may explain the scalar property of multistable perception. Phys Rev Lett 113:1–5. https://doi.org/10.1103/PhysRevLett.113.098103
Chamberland S, Timofeeva Y, Evstratova A et al (2018) Action potential counting at giant mossy fiber terminals gates information transfer in the hippocampus. Proc Natl Acad Sci U S A 115:7434–7439. https://doi.org/10.1073/pnas.1720659115
Chen JL, Penhune VB, Zatorre RJ (2008) Listening to musical rhythms recruits motor regions of the brain. Cereb Cortex 18:2844–2854. https://doi.org/10.1093/cercor/bhn042
Creelman CD (1962) Human Discrimination of Auditory Duration. J Acoust Soc Am 34:582–593. https://doi.org/10.1121/1.1918172
Dehaene S, Changeux JP (1993) Development of elementary numerical abilities: a neuronal model. J Cogn Neurosci 5:390–407. https://doi.org/10.1162/jocn.1993.5.4.390
Egger SW, Le NM, Jazayeri M (2020) A neural circuit model for human sensorimotor timing. Nat Commun 11:1–14. https://doi.org/10.1038/s41467-020-16999-8
Gibbon J (1977) Scalar expectancy theory and Weber’s law in animal timing. Psychol Rev 84:279–325. https://doi.org/10.1037/0033-295X.84.3.279
Gibbon J (1992) Ubiquity of scalar timing with a Poisson clock. J Math Psychol 36:283–293. https://doi.org/10.1016/0022-2496(92)90041-5
Gold JI, Shadlen MN (2007) The neural basis of decision making. Annu Rev Neurosci 30:535–574. https://doi.org/10.1146/annurev.neuro.29.051605.113038
Grahn JA, Brett M (2007) Rhythm and beat perception in motor areas of the brain. J Cogn Neurosci 19:893–906. https://doi.org/10.1162/jocn.2007.19.5.893
Hanes DP, Schall JD (1996) Neural control of voluntary movement initiation. Science 80(274):427–430. https://doi.org/10.1126/science.274.5286.427
Hardy NF, Goudar V, Romero-Sosa JL, Buonomano DV (2018) A model of temporal scaling correctly predicts that motor timing improves with speed. Nat Commun 9:1–14. https://doi.org/10.1038/s41467-018-07161-6
Harrington DL (1998) Temporal processing in the basal ganglia. Neuropsychology 12:3. https://doi.org/10.1037/0894-4105.12.1.3
Haß J, Blaschke S, Rammsayer T, Herrmann JM (2008) A neurocomputational model for optimal temporal processing. J Comput Neurosci 25:449–464. https://doi.org/10.1007/s10827-008-0088-4
Higham DJ (2001) An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev 43:525–546. https://doi.org/10.1137/S0036144500378302
Ivry RB (1996) The representation of temporal information in perception and motor control. Curr Opin Neurobiol 6:851–857. https://doi.org/10.1016/S0959-4388(96)80037-7
Ivry RB, Keele SW (1989) Timing functions of the cerebellum. J Cogn Neurosci 1:136–152. https://doi.org/10.1162/JOCN.1989.1.2.136
Ivry RB, Richardson TC (2002) Temporal control and coordination: the multiple timer model. Brain Cogn 48:117–132. https://doi.org/10.1006/brcg.2001.1308
Karmarkar UR, Buonomano DV (2007) Timing in the absence of clocks: encoding time in neural network states. Neuron 53:427–438. https://doi.org/10.1016/j.neuron.2007.01.006
Killeen PR, Fetterman JG (1988) A Behavioral Theory of Timing. Psychol Rev 95:274–295. https://doi.org/10.1037/0033-295X.95.2.274
Large EW, Almonte FV, Velasco MJ (2010) Author’s personal copy A canonical model for gradient frequency neural networks. Phys D 239:905–911. https://doi.org/10.1016/j.physd.2009.11.015
Leon MI, Shadlen MN (2003) Representation of time by neurons in the posterior parietal cortex of the macaque. Neuron 38:317–327
Long MA, Jin DZ, Fee MS (2010) Support for a synaptic chain model of neuronal sequence generation. Nature 468:394–399. https://doi.org/10.1038/nature09514
Macar F, Lejeune H, Bonnet M et al (2002) Activation of the supplementary motor area and of attentional networks during temporal processing. Exp Brain Res 142:475–485. https://doi.org/10.1007/S00221-001-0953-0
Malapani C, Rakitin B, Levy R et al (1998) Coupled temporal memories in Parkinson’s disease: a dopamine-related dysfunction. J Cogn Neurosci 10:316–331. https://doi.org/10.1162/089892998562762
Martí D, Rinzel J (2013) Dynamics of feature categorization. Neural Comput 25:1–45
Matell MS, Meck WH (2004) Cortico-striatal circuits and interval timing: coincidence detection of oscillatory processes. Cogn Brain Res 21:139–170
Mates J (1994) A model of synchronization of motor acts to a stimulus sequence: I. Timing and error corrections. Biol Cybern 70:463–473. https://doi.org/10.1007/BF00203239
Naud R, Houtman D, Rose GJ, Longtin A (2015) Counting on dis-inhibition: a circuit motif for interval counting and selectivity in the anuran auditory system. J Neurophysiol 114:2804–2815. https://doi.org/10.1152/jn.00138.2015
Nieder A (2005) Counting on neurons: the neurobiology of numerical competence. Nat Rev Neurosci 6:177–190. https://doi.org/10.1038/nrn1626
Nieder A (2016) The neuronal code for number. Nat Rev Neurosci 17:366–382. https://doi.org/10.1038/nrn.2016.40
Nieder A, Dehaene S (2009) Representation of Number in the Brain. Annu Rev Neurosci 32:185–208. https://doi.org/10.1146/annurev.neuro.051508.135550
Paton JJ, Buonomano DV (2018) The neural basis of timing: distributed mechanisms for diverse functions. Neuron. https://doi.org/10.1016/j.neuron.2018.03.045
Rose GJ (2018) The numerical abilities of anurans and their neural correlates: Insights from neuroethological studies of acoustic communication. Philos Trans R Soc B Biol Sci. https://doi.org/10.1098/rstb.2016.0512
Seung HS, Lee DD, Reis BY, Tank DW (2000) Stability of the memory of eye position in a recurrent network of conductance-based model neurons. Neuron 26:259–271. https://doi.org/10.1016/S0896-6273(00)81155-1
Shadlen MN, Newsome WT (1994) Noise, neural codes and cortical organization. Curr Opin Neurobiol 4:569–579. https://doi.org/10.1016/0959-4388(94)90059-0
Simen P, Balci F, deSouza L et al (2011) A model of interval timing by neural integration. J Neurosci 31:9238–9253. https://doi.org/10.1523/JNEUROSCI.3121-10.2011
Teki S, Grube M, Kumar S, Griffiths TD (2011) Distinct neural substrates of duration-based and beat-based auditory timing. J Neurosci 31:3805–3812. https://doi.org/10.1523/JNEUROSCI.5561-10.2011
Treisman M (1963) Temporal discrimination and the indifference interval. Psychol Monogr Gen Appl 77:1–31
Tsodyks MV, Skaggs WE, Sejnowski TJ, Mcnaughton BL (1997) Paradoxical effects of external modulation of inhibitory interneurons. J Neurosci 17:4382–4388
Wallach A, Harvey-Girard E, Jun JJ et al (2018) A time-stamp mechanism may provide temporal information necessary for egocentric to allocentric spatial transformations. Elife 7:1–23. https://doi.org/10.7554/eLife.36769
Wang XJ (2012) Neural dynamics and circuit mechanisms of decision-making. Curr Opin Neurobiol 22:1039–1046. https://doi.org/10.1016/J.CONB.2012.08.006
Wang J, Narain D, Hosseini EA, Jazayeri M (2018) Flexible timing by temporal scaling of cortical responses. Nat Neurosci 21:102–112. https://doi.org/10.1038/s41593-017-0028-6
Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1–24. https://doi.org/10.1016/S0006-3495(72)86068-5
Funding
Nih Blueprint for Neuroscience Research, T90DA043219, Klavdia Zemlianova
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by James Maclaurin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article belongs to S.I. : Stochastic Oscillators.
Rights and permissions
About this article
Cite this article
Zemlianova, K., Bose, A. & Rinzel, J. A biophysical counting mechanism for keeping time. Biol Cybern 116, 205–218 (2022). https://doi.org/10.1007/s00422-021-00915-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00422-021-00915-4