## Abstract

The ability to represent interval timing is crucial for many common behaviors, such as knowing whether to stop when the light turns from green to yellow. Neural representations of interval timing have been reported in the rat primary visual cortex and we have previously presented a computational framework describing how they can be learned by a network of neurons. Recent experimental and theoretical results in entorhinal cortex have shown that single neurons can exhibit persistent activity, previously thought to be generated by a network of neurons. Motivated by these single neuron results, we propose a single spiking neuron model that can learn to compute and represent interval timing. We show that a simple model, reduced analytically to a single dynamical equation, captures the average behavior of the complete high dimensional spiking model very well. Variants of this model can be used to produce bi-stable or multi-stable persistent activity. We also propose a plasticity rule by which this model can learn to represent different intervals and different levels of persistent activity.

### Similar content being viewed by others

## References

Barbieri, F., & Brunel, N. (2008). Can attractor network models account for the statistics of firing during persistent activity in prefrontal cortex?

*Frontiers in Neuroscience, 2*, 114–122.Buhusi, C., & Meck, W. (2002). Differential effects of methamphetamine and haloperidol on the control of an internal clock.

*Behavioral Neuroscience, 116*, 291–297.Compte, A., Constantinidis, C., Tegner, J., Raghavachari, S., Chafee, M., Goldman-Rakic, P., et al. (2003). Temporally irregular mnemonic persistent activity in prefrontal neurons of monkeys during a delayed response task.

*Journal of Neurophysiology, 90*, 3441–3454.Daoudal, G., & Debanne, D. (2003). Long-term plasticity of intrinsic excitability: Learning rules and mechanisms.

*Learning and Memory, 10*(6), 456–465.Durstewitz, D. (2003). Self-organizing neural integrator predicts interval times through climbing activity.

*Journal of Neuroscience, 23*, 5342–5353.Durstewitz, D. (2004). Neural representation of interval time.

*Neuroreport, 15*, 745–749.Egorov, A., Hamam, B., Fransén, E., Hasselmo, M., & Alonso, A. A. (2002). Graded persistent activity in entorhinal cortex neurons.

*Nature, 420*, 173–178.Fransen, E., Alonso, A., & Hasselmo, M. E. (2002). Simulations of the role of the muscarinic-activated calcium-sensitive nonspecific cation current incm in entorhinal neuronal activity during delayed matching tasks.

*Journal of Neuroscience, 22*, 1081–1097.Fransen, E., Tahvildari, B., Egorov, A., Hasselmo, M., & Alonso, A. A. (2006). Mechanism of graded persistent cellular activity of entorhinal cortex layer v neurons.

*Neuron, 49*, 735–746.Gavornik, J., & Shouval, H. (2010). A network of spiking neurons that can represent interval timing: Mean field analysis.

*Journal of Computational Neuroscience*. doi:10.1007/s10827-010-0275-y.Gavornik, J. P., Shuler, M. G., Loewenstein, Y., Bear, M. F., & Shouval, H. Z. (2009). Learning reward timing in cortex through reward dependent expression of synaptic plasticity.

*Proceedings of the National Academy of Sciences of the United States of America, 106*(16), 6826–6831.Goldman-Rakic, P. (1995). Cellular basis of working memory.

*Neuron, 14*, 477–485.Gross, S., Guzma’n, G., Wissenbach, U., Philipp, S., Zhu, M. X., Bruns, D., et al. (2009). TRPC5 is a Ca2-activated channel functionally coupled to Ca2-selective ion channels.

*The Journal of Biological Chemistry, 284*, 34423–34432.Hernandez-Lopez, S., Tkatch, T., Perez-Garci, E., Galarraga, E., Bargas, J., Hamm, H., et al. (2000). D2 dopamine receptors in striatal medium spiny neurons reduce l-type ca2+ currents and excitability via a novel plc[beta]1-ip3-calcineurin-signaling cascade.

*Journal of Neuroscience, 20*, 8987–8995.Johnston, D., & Wu, S.-S. (1994).

*Foundations of cellular neurophysiology*. MIT Press.Loewenstein, Y., & Sompolinsky, H. (2003). Temporal integration by calcium dynamics in a model neuron.

*Nature Neuroscience, 6*, 961–967.Matell, M., Bateson, M., & Meck, W. (2006). Single-trials analyses demonstrate that increases in clock speed contribute to the methamphetamine-inducedhorizontal shifts in peak-interval timing functions.

*Psychopharmacology, 188*, 201–212.Mauk, M., & Buonomano, D. (2004). The neural basis of temporal processing.

*Annual Review of Neuroscience, 27*, 307–340.Meck, W. (1996). Neuropharmacology of timing and time perception.

*Cognitive Brain Research, 3*, 227–242.Saar, D., & Barkai, E. (2003). Long-term modifications in intrinsic neuronal properties and rule learning in rats.

*European Journal of Neuroscience, 17*, 2727–2734.Shalinsky, M., Magistretti, J., Ma, L., & Alonso, A. (2002). Muscarinic activation of a cation current and associated current noise in entorhinal-cortex layer-ii neurons.

*Journal of Neurophysiology, 88,*1197–1211.Shuler, M., & Bear, M. (2006). Reward timing in the primary visual cortex.

*Science, 311*, 1606–1609.Staddon, J., Higa, J., & Chelaru, I. (1999). Time, trace, memory.

*Journal of the Experimental Analysis of Behavior, 71*, 293–301.Surmeier, D. J., Bargas, J., Hemmings, H. J., Nairn, A., & Greengard, P. (1995). Modulation of calcium currents by a d1 dopaminergic protein kinase/phosphatase cascade in rat neostriatal neurons.

*Neuron, 14*, 385–397.Wang, X. (2001). Synaptic reverberation underlying mnemonic persistent activity.

*Trends in Neurosciences, 24*, 455–463.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

**Action Editor: Nicolas Brunel**

## Appendices

### Appendix

### A Calcium dynamics and their analysis

The calcium current in this model arises from high threshold calcium channels, which move to an open state only when the neuron spikes and return to closed rapidly after the end of the action potential (rapidly means much faster that *τ*
_{Ca}).

We simulate these calcium channels using the simple linear expression, instead of the GHK (Johnston and Wu 1994) equation for simplicity:

where *V* is the postsynaptic potential, *E*
_{Ca} the reversal potential for calcium, where *g*
_{Ca} is governed by the dynamical equation:

where *H*
_{Ca} describes the voltage dependence of the calcium channels. Typically we use a hill function with a threshold *θ* and a hill coefficient *m*. However since *τ*
_{gca} < < *τ*
_{Ca} and since voltage changes are rapid as well, the calcium flows in during a very short period, and can be approximated as a step change in calcium levels.

This step is the total amount of calcium transferred through these channels during a single AP, which we define as *ρ*. The parameter *ρ* is a function the calcium channel parameters, such as it’s maximal conductance and it’s time time constant. With this definition and the step approximation Eqs. (1 and 19–20) can be replaced by a single equation of the form:

where *t*
_{
i
} are spike times, and *t*
_{
n
} < *t*. In our simulations we calculate *ρ* from simulations of the detailed calcium dynamics by integrating over the total current generated by a single action potential. We find by comparing the simulations with the detailed dynamics and simulations with the step approximation that this approximation leads only to minor quantitative differences.

If the postsynaptic neuron fires at a fixed frequency *f* with a corresponding inter spike interval Δ*t* = 1/*f*, so that *t*
_{
i
} = *i*Δ*t* we get that:

where \(a=\exp(-\Delta t/\tau_{\rm Ca})\). Defining *t*
^{′} = *t* − *n*Δ*t*, the time since the last spike we get that:

for larger *n* this is approximated by

which can be rewritten as:

where, Ca^{0}(*f*) is the value of calcium shortly after a spike.

By averaging over a single inter-spike interval we get:

If we replace Eq. (21) by the equation:

we obtain at steady state the same expression as in Eq. (27). Equation (28) is the same as Eq. (3).

### B Parameters

Note the parameter *ρ* is calculated inside the code by integrating the calcium dynamics for a single postsynaptic spike and finding the max of the calcium transient. It depends on the parameters *V*
_{theta}, *E*
_{Ca}, *m*, and also on the properties of the action potential. Here we assume that action potential are exponentials with a hight of 100 mV and a width of 1 ms. We measure everything here with respect, to the reversal potential. So the cell voltage relaxes back to zero (rather then approximately − 65 mV). All potentials are therefore measured with respect to this shifted zero.

## Rights and permissions

## About this article

### Cite this article

Shouval, H.Z., Gavornik, J.P. A single spiking neuron that can represent interval timing: analysis, plasticity and multi-stability.
*J Comput Neurosci* **30**, 489–499 (2011). https://doi.org/10.1007/s10827-010-0273-0

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10827-010-0273-0