American Institute of Mathematical Sciences

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

The effect of positive interspike interval correlations on neuronal information transmission

Sven Blankenburg 1, and  Benjamin Lindner 1,

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.
Keywords: information filtering., neural signal transmission, non-renewal point process, Stochastic neuron models.
Mathematics Subject Classification: 93E03, 93E11, 94A12, 94A15, 94A24, 60G10, 60G15, 60G35, 60G50, 60G5.
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:
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L. F. Abbott and W. G. Regehr, Synaptic computation, Nature, 431 (2004), 796-803. doi: 10.1038/nature03010.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

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[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[13]

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

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[41]

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

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[50]

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

[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.  Google Scholar

[52]

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

show all references

References:
[1]

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

[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. Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[41]

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

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[50]

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

[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.  Google Scholar

[52]

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

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