2016, 13(3): 461-481. doi: 10.3934/mbe.2016001

The effect of positive interspike interval correlations on neuronal information transmission

1. 

Bernstein Center for Computational Neuroscience Berlin, Berlin 10115, Germany, Germany

Received  April 2015 Revised  June 2015 Published  January 2016

Experimentally it is known that some neurons encode preferentially information about low-frequency (slow) components of a time-dependent stimulus while others prefer intermediate or high-frequency (fast) components. Accordingly, neurons can be categorized as low-pass, band-pass or high-pass information filters. Mechanisms of information filtering at the cellular and the network levels have been suggested. Here we propose yet another mechanism, based on noise shaping due to spontaneous non-renewal spiking statistics. We compare two integrate-and-fire models with threshold noise that differ solely in their interspike interval (ISI) correlations: the renewal model generates independent ISIs, whereas the non-renewal model exhibits positive correlations between adjacent ISIs. For these simplified neuron models we analytically calculate ISI density and power spectrum of the spontaneous spike train as well as approximations for input-output cross-spectrum and spike-train power spectrum in the presence of a broad-band Gaussian stimulus. This yields the spectral coherence, an approximate frequency-resolved measure of information transmission. We demonstrate that for low spiking variability the renewal model acts as a low-pass filter of information (coherence has a global maximum at zero frequency), whereas the non-renewal model displays a pronounced maximum of the coherence at non-vanishing frequency and thus can be regarded as a band-pass filter of information.
Citation: Sven Blankenburg, Benjamin Lindner. The effect of positive interspike interval correlations on neuronal information transmission. Mathematical Biosciences & Engineering, 2016, 13 (3) : 461-481. doi: 10.3934/mbe.2016001
References:
[1]

L. F. Abbott and W. G. Regehr, Synaptic computation, Nature, 431 (2004), 796-803. doi: 10.1038/nature03010.

[2]

R. Azouz and C. M. Gray, Dynamic spike threshold reveals a mechanism for synaptic coincidence detection in cortical neurons in vivo, Proc. Nat. Acad. Sci., 97 (2000), 8110-8115.

[3]

D. Bernardi and B. Lindner, A frequency-resolved mutual information rate and its application to neural systems, J. Neurophysiol., 113 (2014), 1342-1357. doi: 10.1152/jn.00354.2014.

[4]

S. Blankenburg, W. Wu, B. Lindner and S. Schreiber, Information filtering in resonant neurons, J. Comput. Neurosci., 39 (2015), 349-370.

[5]

A. Borst and F. Theunissen, Information theory and neural coding, Nat. Neurosci., 2 (1999), 947-957. doi: 10.1038/14731.

[6]

N. Brenner, O. Agam, W. Bialek and R. de Ruyter van Steveninck, Statistical properties of spike trains: Universal and stimulus-dependent aspects, Phys. Rev. E, 66 (2002), 031907, 14pp. doi: 10.1103/PhysRevE.66.031907.

[7]

P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, Springer, 2009.

[8]

N. Brunel, V. Hakim and M. J. E. Richardson, Firing-rate resonance in a generalized integrate-and-fire neuron with subthreshold resonance, Phys. Rev. E, 67 (2003), 051916, 23pp. doi: 10.1103/PhysRevE.67.051916.

[9]

M. Chacron, B. Lindner and A. Longtin, Threshold fatigue and information transfer, J. Comput. Neurosci., 23 (2007), 301-311. doi: 10.1007/s10827-007-0033-y.

[10]

M. J. Chacron, A. Longtin and L. Maler, Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli, J. Neurosci., 21 (2001), 5328-5343.

[11]

M. J. Chacron, B. Doiron, L. Maler, A. Longtin and J. Bastian, Non-classical receptive field mediates switch in a sensory neuron's frequency tuning, Nature, 423 (2003), 77-81.

[12]

M. J. Chacron, B. Lindner and A. Longtin, Noise shaping by interval correlations increases information transfer, Phys. Rev. Lett., 93 (2004), 059904.

[13]

T. Cover and J. Thomas, Elements of Information Theory, Wiley, New-York, 1991. doi: 10.1002/0471200611.

[14]

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events, Chapman and Hall, London, 1966.

[15]

F. Droste, T. Schwalger and B. Lindner, Interplay of two signals in a neuron with short-term synaptic plasticity, Front. Comp. Neurosci., 7 (2013), p86. doi: 10.3389/fncom.2013.00086.

[16]

T. A. Engel, L. Schimansky-Geier, A. V. M. Herz, S. Schreiber and I. Erchova, Subthreshold membrane-potential resonances shape spike-train patterns in the entorhinal cortex, J. Neurophysiol., 100 (2008), 1576-1589. doi: 10.1152/jn.01282.2007.

[17]

K. Fisch, T. Schwalger, B. Lindner, A. Herz and J. Benda, Channel noise from both slow adaptation currents and fast currents is required to explain spike-response variability in a sensory neuron, J. Neurosci., 32 (2012), 17332-17344. doi: 10.1523/JNEUROSCI.6231-11.2012.

[18]

J. L. Folks and R. S. Chhikara, The inverse gaussian distribution and its statistical application-a review, J. R. Statist. Soc. B, 40 (1978), 263-289.

[19]

F. Gabbiani, Coding of time-varying signals in spike trains of linear and half-wave rectifying neurons, Network Comp. Neural., 7 (1996), 61-85.

[20]

C. D. Geisler and J. M. Goldberg, A stochastic model of repetitive activity of neurons, Biophys. J., 6 (1966), 53-69.

[21]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys. J., 4 (1964), 41-68.

[22]

W. Gerstner and W. M. Kistler, Spiking Neuron Models, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511815706.

[23]

J. D. Hamilton, Time Series Analysis, Princeton University Press, 1994.

[24]

A. V. Holden, Models of the Stochastic Activity of Neurones, Springer-Verlag, Berlin, 1976.

[25]

E. M. Izhikevich, Resonate-and-fire neurons, Neural. Netw., 14 (2001), 883-894.

[26]

B. Lindner, Interspike interval statistics of neurons driven by colored noise, Phys. Rev. E, 69 (2004), 022901.

[27]

B. Lindner, Low-pass filtering of information in the leaky integrate-and-fire neuron driven by white noise, in International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012) (eds. I. Visarath, A. Palacios and P. Longhini), Springer, 2012.

[28]

B. Lindner, M. J. Chacron, A. Longtin, Integrate-and-fire neurons with threshold noise - a tractable model of how interspike interval correlations affect neuronal signal transmission, Phys. Rev. E, 72 (2005), p021911, 21pp. doi: 10.1103/PhysRevE.72.021911.

[29]

B. Lindner, D. Gangloff, A. Longtin and J. E. Lewis, Broadband coding with dynamic synapses, J. Neurosci., 29 (2009), 2076-2087. doi: 10.1523/JNEUROSCI.3702-08.2009.

[30]

S. B. Lowen and M. C. Teich, Auditory-nerve action potentials form a nonrenewal point process over short as well as long time scales, J. Acoust. Soc. Am., 92 (1992), 803-806.

[31]

D. J. Mar, C. C. Chow, W. Gerstner, R. W. Adams and J. J. Collins, Noise shaping in populations of coupled model neurons, Proc. Natl. Acad. Sci., 96 (1999), 10450-10455. doi: 10.1073/pnas.96.18.10450.

[32]

G. Marsat and G. S. Pollack, Differential temporal coding of rhythmically diverse acoustic signals by a single interneuron, J. Neurophysiol., 92 (2004), 939-948.

[33]

C. Massot, M. Chacron and K. Cullen, Information transmission and detection thresholds in the vestibular nuclei: Single neurons vs. population encoding, J. Neurophysiol., 105 (2011), 1798-1814. doi: 10.1152/jn.00910.2010.

[34]

M. Merkel and B. Lindner, Synaptic filtering of rate-coded information, Phys. Rev. E, 81 (2010), 041921, 19pp. doi: 10.1103/PhysRevE.81.041921.

[35]

J. W. Middleton, A. Longtin, J. Benda and L. Maler, Postsynaptic receptive field size and spike threshold determine encoding of high-frequency information via sensitivity to synchronous presynaptic activity, J. Neurophysiol., 101 (2009), 1160-1170. doi: 10.1152/jn.90814.2008.

[36]

A. B. Neiman and D. F. Russell, Sensory coding in oscillatory electroreceptors of paddlefish, Chaos, 21 (2011), 047505. doi: 10.1063/1.3669494.

[37]

A. Nikitin, N. Stocks and A. Bulsara, Enhancing the resolution of a sensor via negative correlation: A biologically inspired approach, Phys. Rev. Lett., 109 (2012), 238103.

[38]

A. M. M. Oswald, M. J. Chacron, B. Doiron, J. Bastian and L. Maler, Parallel processing of sensory input by bursts and isolated spikes, J. Neurosci., 24 (2004), 4351-4362. doi: 10.1523/JNEUROSCI.0459-04.2004.

[39]

S. A. Prescott and T. J. Sejnowski, Spike-rate coding and spike-time coding are affected oppositely by different adaptation mechanisms, J. Neurosci., 28 (2008), 13649-13661. doi: 10.1523/JNEUROSCI.1792-08.2008.

[40]

F. Rieke, D. Bodnar and W. Bialek, Naturalistic stimuli increase the rate and efficiency of information transmission by primary auditory afferents, Proc. Biol. Sci., 262 (1995), 259-265.

[41]

F. Rieke, D. Warland, R. de Ruyter van Steveninck and W. Bialek, Spikes: Exploring the Neural Code, MIT Press, Cambridge, Massachusetts, 1999.

[42]

J. C. Roddey, B. Girish and J. P. Miller, Assessing the performance of neural encoding models in the presence of noise, J. Comput. Neurosci., 8 (2000), 95-112.

[43]

S. G. Sadeghi, M. J. Chacron, M. C. Taylor and K. E. Cullen, Neural variability, detection thresholds, and information transmission in the vestibular system, J. Neurosci., 27 (2007), 771-781.

[44]

T. Schwalger, K. Fisch, J. Benda and B. Lindner, How noisy adaptation of neurons shapes interspike interval histograms and correlations, PLoS Comp. Biol., 6 (2010), e1001026, 25pp. doi: 10.1371/journal.pcbi.1001026.

[45]

R. Shannon, The mathematical theory of communication, Bell Syst. Tech. J., 27 (1948), 379-423. doi: 10.1002/j.1538-7305.1948.tb01338.x.

[46]

N. Sharafi, J. Benda and B. Lindner, Information filtering by synchronous spikes in a neural population, J. Comp. Neurosci., 34 (2013), 285-301. doi: 10.1007/s10827-012-0421-9.

[47]

L. Shiau, T. Schwalger and B. Lindner, ISI correlation in a stochastic exponential integrate-and-fire model with subthreshold and spike-triggered adaptation, J. Comp. Neurosci., 38 (2015), 589-600. doi: 10.1007/s10827-015-0558-4.

[48]

J. Shin, The noise shaping neural coding hypothesis: A brief history and physiological implications, Neurocomp., 44 (2002), 167-175. doi: 10.1016/S0925-2312(02)00379-X.

[49]

J. H. Shin, K. R. Lee and S. B. Park, Novel neural circuits based on stochastic pulse coding and noise feedback pulse coding, Int. J. Electronics, 74 (1993), 359-368. doi: 10.1080/00207219308925840.

[50]

R. L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, New York, 1967.

[51]

R. D. Vilela and B. Lindner, Comparative study of different integrate-and-fire neurons: Spontaneous activity, dynamical response, and stimulus-induced correlation, Phys. Rev. E, 80 (2009), 031909. doi: 10.1103/PhysRevE.80.031909.

[52]

R. S. Zucker and W. G. Regehr, Short-term synaptic plasticity, Ann. Rev. Physiol., 64 (2002), 355-405.

show all references

References:
[1]

L. F. Abbott and W. G. Regehr, Synaptic computation, Nature, 431 (2004), 796-803. doi: 10.1038/nature03010.

[2]

R. Azouz and C. M. Gray, Dynamic spike threshold reveals a mechanism for synaptic coincidence detection in cortical neurons in vivo, Proc. Nat. Acad. Sci., 97 (2000), 8110-8115.

[3]

D. Bernardi and B. Lindner, A frequency-resolved mutual information rate and its application to neural systems, J. Neurophysiol., 113 (2014), 1342-1357. doi: 10.1152/jn.00354.2014.

[4]

S. Blankenburg, W. Wu, B. Lindner and S. Schreiber, Information filtering in resonant neurons, J. Comput. Neurosci., 39 (2015), 349-370.

[5]

A. Borst and F. Theunissen, Information theory and neural coding, Nat. Neurosci., 2 (1999), 947-957. doi: 10.1038/14731.

[6]

N. Brenner, O. Agam, W. Bialek and R. de Ruyter van Steveninck, Statistical properties of spike trains: Universal and stimulus-dependent aspects, Phys. Rev. E, 66 (2002), 031907, 14pp. doi: 10.1103/PhysRevE.66.031907.

[7]

P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, Springer, 2009.

[8]

N. Brunel, V. Hakim and M. J. E. Richardson, Firing-rate resonance in a generalized integrate-and-fire neuron with subthreshold resonance, Phys. Rev. E, 67 (2003), 051916, 23pp. doi: 10.1103/PhysRevE.67.051916.

[9]

M. Chacron, B. Lindner and A. Longtin, Threshold fatigue and information transfer, J. Comput. Neurosci., 23 (2007), 301-311. doi: 10.1007/s10827-007-0033-y.

[10]

M. J. Chacron, A. Longtin and L. Maler, Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli, J. Neurosci., 21 (2001), 5328-5343.

[11]

M. J. Chacron, B. Doiron, L. Maler, A. Longtin and J. Bastian, Non-classical receptive field mediates switch in a sensory neuron's frequency tuning, Nature, 423 (2003), 77-81.

[12]

M. J. Chacron, B. Lindner and A. Longtin, Noise shaping by interval correlations increases information transfer, Phys. Rev. Lett., 93 (2004), 059904.

[13]

T. Cover and J. Thomas, Elements of Information Theory, Wiley, New-York, 1991. doi: 10.1002/0471200611.

[14]

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events, Chapman and Hall, London, 1966.

[15]

F. Droste, T. Schwalger and B. Lindner, Interplay of two signals in a neuron with short-term synaptic plasticity, Front. Comp. Neurosci., 7 (2013), p86. doi: 10.3389/fncom.2013.00086.

[16]

T. A. Engel, L. Schimansky-Geier, A. V. M. Herz, S. Schreiber and I. Erchova, Subthreshold membrane-potential resonances shape spike-train patterns in the entorhinal cortex, J. Neurophysiol., 100 (2008), 1576-1589. doi: 10.1152/jn.01282.2007.

[17]

K. Fisch, T. Schwalger, B. Lindner, A. Herz and J. Benda, Channel noise from both slow adaptation currents and fast currents is required to explain spike-response variability in a sensory neuron, J. Neurosci., 32 (2012), 17332-17344. doi: 10.1523/JNEUROSCI.6231-11.2012.

[18]

J. L. Folks and R. S. Chhikara, The inverse gaussian distribution and its statistical application-a review, J. R. Statist. Soc. B, 40 (1978), 263-289.

[19]

F. Gabbiani, Coding of time-varying signals in spike trains of linear and half-wave rectifying neurons, Network Comp. Neural., 7 (1996), 61-85.

[20]

C. D. Geisler and J. M. Goldberg, A stochastic model of repetitive activity of neurons, Biophys. J., 6 (1966), 53-69.

[21]

G. L. Gerstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys. J., 4 (1964), 41-68.

[22]

W. Gerstner and W. M. Kistler, Spiking Neuron Models, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511815706.

[23]

J. D. Hamilton, Time Series Analysis, Princeton University Press, 1994.

[24]

A. V. Holden, Models of the Stochastic Activity of Neurones, Springer-Verlag, Berlin, 1976.

[25]

E. M. Izhikevich, Resonate-and-fire neurons, Neural. Netw., 14 (2001), 883-894.

[26]

B. Lindner, Interspike interval statistics of neurons driven by colored noise, Phys. Rev. E, 69 (2004), 022901.

[27]

B. Lindner, Low-pass filtering of information in the leaky integrate-and-fire neuron driven by white noise, in International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012) (eds. I. Visarath, A. Palacios and P. Longhini), Springer, 2012.

[28]

B. Lindner, M. J. Chacron, A. Longtin, Integrate-and-fire neurons with threshold noise - a tractable model of how interspike interval correlations affect neuronal signal transmission, Phys. Rev. E, 72 (2005), p021911, 21pp. doi: 10.1103/PhysRevE.72.021911.

[29]

B. Lindner, D. Gangloff, A. Longtin and J. E. Lewis, Broadband coding with dynamic synapses, J. Neurosci., 29 (2009), 2076-2087. doi: 10.1523/JNEUROSCI.3702-08.2009.

[30]

S. B. Lowen and M. C. Teich, Auditory-nerve action potentials form a nonrenewal point process over short as well as long time scales, J. Acoust. Soc. Am., 92 (1992), 803-806.

[31]

D. J. Mar, C. C. Chow, W. Gerstner, R. W. Adams and J. J. Collins, Noise shaping in populations of coupled model neurons, Proc. Natl. Acad. Sci., 96 (1999), 10450-10455. doi: 10.1073/pnas.96.18.10450.

[32]

G. Marsat and G. S. Pollack, Differential temporal coding of rhythmically diverse acoustic signals by a single interneuron, J. Neurophysiol., 92 (2004), 939-948.

[33]

C. Massot, M. Chacron and K. Cullen, Information transmission and detection thresholds in the vestibular nuclei: Single neurons vs. population encoding, J. Neurophysiol., 105 (2011), 1798-1814. doi: 10.1152/jn.00910.2010.

[34]

M. Merkel and B. Lindner, Synaptic filtering of rate-coded information, Phys. Rev. E, 81 (2010), 041921, 19pp. doi: 10.1103/PhysRevE.81.041921.

[35]

J. W. Middleton, A. Longtin, J. Benda and L. Maler, Postsynaptic receptive field size and spike threshold determine encoding of high-frequency information via sensitivity to synchronous presynaptic activity, J. Neurophysiol., 101 (2009), 1160-1170. doi: 10.1152/jn.90814.2008.

[36]

A. B. Neiman and D. F. Russell, Sensory coding in oscillatory electroreceptors of paddlefish, Chaos, 21 (2011), 047505. doi: 10.1063/1.3669494.

[37]

A. Nikitin, N. Stocks and A. Bulsara, Enhancing the resolution of a sensor via negative correlation: A biologically inspired approach, Phys. Rev. Lett., 109 (2012), 238103.

[38]

A. M. M. Oswald, M. J. Chacron, B. Doiron, J. Bastian and L. Maler, Parallel processing of sensory input by bursts and isolated spikes, J. Neurosci., 24 (2004), 4351-4362. doi: 10.1523/JNEUROSCI.0459-04.2004.

[39]

S. A. Prescott and T. J. Sejnowski, Spike-rate coding and spike-time coding are affected oppositely by different adaptation mechanisms, J. Neurosci., 28 (2008), 13649-13661. doi: 10.1523/JNEUROSCI.1792-08.2008.

[40]

F. Rieke, D. Bodnar and W. Bialek, Naturalistic stimuli increase the rate and efficiency of information transmission by primary auditory afferents, Proc. Biol. Sci., 262 (1995), 259-265.

[41]

F. Rieke, D. Warland, R. de Ruyter van Steveninck and W. Bialek, Spikes: Exploring the Neural Code, MIT Press, Cambridge, Massachusetts, 1999.

[42]

J. C. Roddey, B. Girish and J. P. Miller, Assessing the performance of neural encoding models in the presence of noise, J. Comput. Neurosci., 8 (2000), 95-112.

[43]

S. G. Sadeghi, M. J. Chacron, M. C. Taylor and K. E. Cullen, Neural variability, detection thresholds, and information transmission in the vestibular system, J. Neurosci., 27 (2007), 771-781.

[44]

T. Schwalger, K. Fisch, J. Benda and B. Lindner, How noisy adaptation of neurons shapes interspike interval histograms and correlations, PLoS Comp. Biol., 6 (2010), e1001026, 25pp. doi: 10.1371/journal.pcbi.1001026.

[45]

R. Shannon, The mathematical theory of communication, Bell Syst. Tech. J., 27 (1948), 379-423. doi: 10.1002/j.1538-7305.1948.tb01338.x.

[46]

N. Sharafi, J. Benda and B. Lindner, Information filtering by synchronous spikes in a neural population, J. Comp. Neurosci., 34 (2013), 285-301. doi: 10.1007/s10827-012-0421-9.

[47]

L. Shiau, T. Schwalger and B. Lindner, ISI correlation in a stochastic exponential integrate-and-fire model with subthreshold and spike-triggered adaptation, J. Comp. Neurosci., 38 (2015), 589-600. doi: 10.1007/s10827-015-0558-4.

[48]

J. Shin, The noise shaping neural coding hypothesis: A brief history and physiological implications, Neurocomp., 44 (2002), 167-175. doi: 10.1016/S0925-2312(02)00379-X.

[49]

J. H. Shin, K. R. Lee and S. B. Park, Novel neural circuits based on stochastic pulse coding and noise feedback pulse coding, Int. J. Electronics, 74 (1993), 359-368. doi: 10.1080/00207219308925840.

[50]

R. L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, New York, 1967.

[51]

R. D. Vilela and B. Lindner, Comparative study of different integrate-and-fire neurons: Spontaneous activity, dynamical response, and stimulus-induced correlation, Phys. Rev. E, 80 (2009), 031909. doi: 10.1103/PhysRevE.80.031909.

[52]

R. S. Zucker and W. G. Regehr, Short-term synaptic plasticity, Ann. Rev. Physiol., 64 (2002), 355-405.

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