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Oscillations and variability in neuronal systems: interplay of autonomous transient dynamics and fast deterministic fluctuations

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Abstract

Neuronal systems are subject to rapid fluctuations both intrinsically and externally. These fluctuations can be disruptive or constructive. We investigate the dynamic mechanisms underlying the interactions between rapidly fluctuating signals and the intrinsic properties of the target cells to produce variable and/or coherent responses. We use linearized and non-linear conductance-based models and piecewise constant (PWC) inputs with short duration pieces. The amplitude distributions of the constant pieces consist of arbitrary permutations of a baseline PWC function. In each trial within a given protocol we use one of these permutations and each protocol consists of a subset of all possible permutations, which is the only source of uncertainty in the protocol. We show that sustained oscillatory behavior can be generated in response to various forms of PWC inputs independently of whether the stable equilibria of the corresponding unperturbed systems are foci or nodes. The oscillatory voltage responses are amplified by the model nonlinearities and attenuated for conductance-based PWC inputs as compared to current-based PWC inputs, consistent with previous theoretical and experimental work. In addition, the voltage responses to PWC inputs exhibited variability across trials, which is reminiscent of the variability generated by stochastic noise (e.g., Gaussian white noise). Our analysis demonstrates that both oscillations and variability are the result of the interaction between the PWC input and the target cell’s autonomous transient dynamics with little to no contribution from the dynamics in vicinities of the steady-state, and do not require input stochasticity.

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References

  • Allen, E. J., Novosel, S. J., & Zhang, Z. (1998). Finite element and difference approximation of some linear stochastic partial differential equations. Stochastics and Stochastic Reports, 64, 117–142.

    Article  Google Scholar 

  • Amit, D. J. (1989). Modeling brain function: The world of attractor neural networks. New York, NY: Cambridge University Press.

    Book  Google Scholar 

  • Anishchenko, V. S., & Neiman, B. (1997). Stochastic synchronization. Stochastic Dynamics (in Lecture Notes Physics), 484, 154–166.

    Google Scholar 

  • Arieli, A., Sterkin, A., Grinvald, A., & Aertsen, A. D. (1996). Dynamics of ongoing activity: explanation of the large variability in evoked cortical responses. Science, 273(5283), 1868–1871.

    Article  CAS  PubMed  Google Scholar 

  • Baltanas, J. P., & Casado, J. M. (1998). Bursting behavior of the Fitzhugh-Nagumo neuron model subject to quasi-monochromatic noise. Physica D: Nonlinear Phenomena, 122, 231–240.

    Article  Google Scholar 

  • Baspinar, E., Schulen, L., Olmi, S., & Zakharova, A. (2021). Coherence resonance in neuronal populations: Mean-field versus network model. Physical Review E, 103, 032308.

    Article  CAS  PubMed  Google Scholar 

  • Benzi, R., Parisi, G., Sutera, A., & Vulpiani, A. (1982). Stochastic resonance in climatic change. Tellus, 34, 10–16.

    Article  Google Scholar 

  • Bernstein, J. G., & Boyden, E. S. (2012). Optogenetic tools for analyzing the neural circuits of behavior. Current Opinion in Neurobiology, 22, 61–71.

    Article  CAS  PubMed  Google Scholar 

  • Bondanelli, G., & Ostojic, S. (2020). Coding with transient trajectories in recurrent neural networks. PLoS Computational Biology, 16, e1007655.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Boucheny, C., Brunel, N., & Arleo, A. (2005). A continuous attractor network model without recurrent excitation: maintenance and integration in the head direction cell system. Journal of Computational Neuroscience, 18, 205–227.

    Article  PubMed  Google Scholar 

  • Britten, K. H., Shadlen, M. N., Newsome, W. T., & Movshon, J. A. (1992). The analysis of visual motion: a comparison of neuronal and psychophysical performance. Journal of Neuroscience, 12(12), 4745–4765.

    Article  CAS  PubMed  Google Scholar 

  • Burden, R. L., & Faires, J. D. (1980). Numerical analysis. PWS Publishing Company - Boston.

  • Brunel, N., Chance, F. S., Fourcaud, N., & Abbott, L. F. (2001). Effects of synaptic noise and filtering on the frequency response of spiking neurons. Physical Review Letters, 86, 2186–2189.

    Article  CAS  PubMed  Google Scholar 

  • Calvin, W. H., & Stevens, C. F. (1967). Synaptic noise as a source of variability in the interval between action potentials. Science, 155, 842–844.

    Article  CAS  PubMed  Google Scholar 

  • Chow, C. C., & White, J. A. (1996). Spontaneous action potentials due to channel fluctuation. Biophysical Journal, 71, 3013–3021.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Churchland, M. M., Byron, M. Y., Ryu, S. I., Santhanam, G., & Shenoy, K. V. (2006). Neural variability in premotor cortex provides a signature of motor preparation. Journal of Neuroscience, 26(14), 3697–3712.

    Article  CAS  PubMed  Google Scholar 

  • Churchland, M. M., Yu, B. M., Cunningham, J. P., Sugrue, L. P., Cohen, M. R., Corrado, G. S., et al. (2010). Stimulus onset quenches neural variability: a widespread cortical phenomenon. Nature Neuroscience, 13, 369–378.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Cohen, M. R., & Kohn, A. (2011). Measuring and interpreting neuronal correlations. Nature Neuroscience, 14(7), 811–819.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Collins, J. J., Chow, C. C., & Imhoff, T. T. (1995). Aperiodic stochastic resonance in excitable systems. Physical Review E, 76, 642–645.

    Google Scholar 

  • Collins, J. J., Imhoff, T. T., & Grigg, P. (1996). Noise-enhanced information transmission in rat SA1 cutaneous mechanoreceptors via aperiodic stochastic resonance. Journal of Neurophysiology, 76, 642–645.

    Article  CAS  PubMed  Google Scholar 

  • Day, J., Rubin, J. E., & Chow, C. C. (2009). Competition between transients in the rate of approach to a fixed point. SIAM Journal on Applied Dynamical Systems, 8(4), 1523–1563.

    Article  PubMed  PubMed Central  Google Scholar 

  • Deco, G., Rolls, E., & Romo, R. (2009). Stochastic dynamics as a principle of brain function. Progress in Neurobiology, 88, 1–16.

    Article  PubMed  Google Scholar 

  • DeFelice, L. J. (1981). Introduction to channel noise. Plenum Press.

    Google Scholar 

  • Deisseroth, K. (2011). Optogenetics. Nature Methods, 8, 26–29.

    Article  CAS  PubMed  Google Scholar 

  • Destexhe, A., Badoual, M., Piwkowska, Z., Bal, T., & Rudolph, M. (2004). A novel method for characterizing synaptic noise in cortical neurons. Neurocomputing, 58, 191–196.

    Article  Google Scholar 

  • Destexhe, A., & Rudolph-Lilith, M. (2012). Neuronal Noise. Springer.

    Book  Google Scholar 

  • DeVille, R. L., Vanden-Eijnden, E., & Muratov, C. B. (2005). Two distinct mechanisms of coherence in randomly perturbed dynamical systems. Physical Review E, 72, 031105.

    Article  CAS  Google Scholar 

  • Dorval, A. D., Jr., & White, J. A. (2005). Channel noise is essential for perithreshold oscillations in entorhinal stellate neurons. Journal of Neuroscience, 25, 10025–10028.

    Article  CAS  PubMed  Google Scholar 

  • Douglass, J. K., Wilkens, L., Pantazelou, E., & Moss, F. (1993). Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance. Nature, 365, 337–340.

    Article  CAS  PubMed  Google Scholar 

  • Du, Q., & Zhang, T. (2002). Numerical approximation of some linear stochastic partial differential equations driven by special additive noise. SIAM Journal on Numerical Analysis, 400, 1421–1445.

    Article  Google Scholar 

  • Faisal, A. A., Selen, L. P., & Wolpert, D. M. (2008). Noise in the nervous system. Nature Reviews Neuroscience, 9(4), 292–303.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Fellous, J. M., Rudolph, M., Destexhe, A., & Sejnowski, T. J. (2003). Synaptic background noise controls the input/output characteristics of single cells in an in vitro model of in vivo activity. Neuroscience, 122, 811–829.

    Article  CAS  PubMed  Google Scholar 

  • Fernandez, R., & White, J. A. (2008). Artificial synaptic conductances reduce subthreshold oscillations and periodic firing in stellate cells of the entorhinal cortex. Journal of Neuroscience, 28, 3790–3803.

    Article  CAS  PubMed  Google Scholar 

  • Fox, M. D., Snyder, A. Z., Zacks, J. M., & Raichle, M. E. (2006). Coherent spontaneous activity accounts for trial-to-trial variability in human evoked brain responses. Nature Neuroscience, 9(1), 23–25.

    Article  CAS  PubMed  Google Scholar 

  • Gammaitoni, L., Hanggi, P., Jung, P., & Marchesoni, F. (1998). Stochastic resonance. Reviews of Modern Physics, 70, 223–287.

    Article  CAS  Google Scholar 

  • Gardiner, C. W. (1985). Handbook of Stochastic Methods. Berlin: Springer-Verlag.

    Google Scholar 

  • Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer-Verlag.

    Book  Google Scholar 

  • Hakim, V., & Rappel, W.-J. (1994). Noise-induced periodic behaviour in the globally coupled complex Ginzburg-Landau equation. Europhysics Letters, 27, 637–642.

    Article  Google Scholar 

  • Hong, S., Ratté, S., Prescott, S. A., & De Schutter, E. (2012). Single neuron firing properties impact correlation-based population coding. Journal of Neuroscience, 32, 1413–1428.

    Article  CAS  PubMed  Google Scholar 

  • Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National academy of Sciences of the United States of America, 79, 2554–2558.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Hutcheon, B., Miura, R. M., & Puil, E. (1996). Subthreshold membrane resonance in neocortical neurons. Journal of Neurophysiology, 76, 683–697.

    Article  CAS  PubMed  Google Scholar 

  • Hutcheon, B., & Yarom, Y. (2000). Resonance, oscillations and the intrinsic frequency preferences in neurons. Trends in Neurosciences, 23, 216–222.

    Article  CAS  PubMed  Google Scholar 

  • Ito, T., Brincat, S. L., Mil, R. D., Siegel, M., He, B. J., Miller, E. K., et al. (2020). Task-evoked activity quenches neural correlations and variability in large-scale brain systems. PLoS Computational Biology, 16, e1007983.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Izhikevich, E. (2006). Dynamical Systems in Neuroscience: The geometry of excitability and bursting. MIT Press (Cambridge, Massachusetts).

  • Jensen, R. V. (1998). Synchronization of randomly driven nonlinear oscillators. Physical Review E, 58, R6907–R6910.

    Article  CAS  Google Scholar 

  • Knierim, J. J., & Zhang, K. (2012). Attractor dynamics of spatially correlated neural activity in the limbic system. Annual Review of Neuroscience, 32, 267–285.

    Article  CAS  Google Scholar 

  • Kurrer, C., & Schulten, K. (1995). Noise-induced synchronous neuronal oscillations. Physical Review E, 51, 6213–6218.

    Article  CAS  Google Scholar 

  • Laing, C., & Lord, G. J. (2010). Stochastic methods in neuroscience. Oxford University Press.

    Google Scholar 

  • Lee, J., & Lee, J. (2018). Quantitative analysis of a transient dynamics of a gene regulatory network. Physical Review E, 98, 062404.

    Article  CAS  Google Scholar 

  • Lee, S.-G., Neiman, A., & Kim, S. (1998). Coherence resonance in a hodgkin-huxley neuron. Physical Review E, 57, 3292–3297.

    Article  CAS  Google Scholar 

  • Levenstein, D., Buzsaki, G., & Rinzel, J. (2019). Nrem sleep in the rodent neocortex and hippocampus reflects excitable dynamics. Nature Communications, 10, 2478.

    Article  PubMed  PubMed Central  Google Scholar 

  • Lim, S., & Rinzel, J. (2010). Noise-induced transitions in slow wave neuronal dynamics. Journal of Computational Neuroscience, 28, 1–17.

    Article  PubMed  Google Scholar 

  • Lindner, B., García-Ojalvo, J., Neiman, A., & Schimansky-Geier, L. (2004). Effects of noise in excitable systems. Physics Reports, 392, 321–424.

    Article  Google Scholar 

  • Longtin, A., Bulsara, A., & Moss, F. (1991). Time-interval sequences in bistable systems and the noise-induced transmission of information by sensory neurons. Physical Review Letters, 67, 656.

    Article  CAS  PubMed  Google Scholar 

  • Marin, B., Pinto, R. D., Elson, R. C., & Colli, E. (2014). Noise, transient dynamics, and the generation of realistic interspike interval variation in square-wave burster neurons. Physical Review E, 90, 042718.

    Article  CAS  Google Scholar 

  • Mato, G. (1989). Stochastic resonance using noise generated by a neural network. Physical Review E, 59, 3339–3343.

    Article  Google Scholar 

  • Matsumoto, K., & Tsuda, I. (1983). Noise-induced order. Journal of Statistical Physics, 31, 87–106.

    Article  Google Scholar 

  • Mazor, O., & Laurent, G. (2005). Transient dynamics versus fixed points in odor representations by locust antennal lobe projection neurons. Neuron, 48, 661–673.

    Article  CAS  PubMed  Google Scholar 

  • McDonnell, M. D., & Abboott, D. (2009). What is stochastic resonance? definitions, misconceptions, debates, and its relevance to biology. PLoS Computational Biology, 5, e1000348.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • McNamara, B., & Wiesenfeld, K. (1989). Theory of stochastic resonance. Physical Review A, 39, 4854–4869.

    Article  CAS  Google Scholar 

  • Middleton, J., Chacron, M., Lindner, B., & Longtin, A. (2003). Firing statistics of a neuron model driven by long-range correlated noise. Physical Review E, 68, 021920.

    Article  CAS  Google Scholar 

  • Muratov, C. B., Vanden-Eijnden, E., & Weinan, E. (2005). Self-induced stochastic resonance in excitable systems. Physica D: Nonlinear Phenomena, 210, 227–240.

    Article  CAS  Google Scholar 

  • Nachstedt, T., & Tetzlaff, C. (2016). Working memory requires a combination of transient and attractor-dominated dynamics to process unreliably timed inputs. Science and Reports, 7, 2473.

    Article  CAS  Google Scholar 

  • Neiman, A., Saparin, P. I., & Stone, L. (1997). Coherence resonance at noisy precursors of bifurcations in nonlinear dynamical systems. Physical Review E, 56, 270–273.

    Article  CAS  Google Scholar 

  • Pena, R. F. O., & Rotstein, H. G. (2022). The voltage and spiking responses of subthreshold resonant neurons to structured and fluctuating inputs: persistence and loss of resonance and variability. Biological Cybernetics, 116, 163-190.

    Article  PubMed  Google Scholar 

  • Pena, R. F. O., Zaks, M. A., & Roque, A. C. (2018). Dynamics of spontaneous activity in random networks with multiple neuron subtypes and synaptic noise. Journal of Computational Neuroscience, 45, 1–28.

    Article  PubMed  PubMed Central  Google Scholar 

  • Pham, J., Pakdaman, K., & Vibert, J.-F. (1998). Noise-induced coherent oscillations in randomly connected neural networks. Physical Review E, 58, 3610–3622.

    Article  CAS  Google Scholar 

  • Pikovsky, A. S. (1984). Synchronization and stochastization of nonlinear oscillations by external noise. In: Nonlinear and Turbulent Processes in Physics, ed. Sagdeev, R. Z. Harwood Acad. Publ., 3:1601–1604.

  • Pikovsky, A. S., & Kurths, J. (1997). Coherence resonance in a noise-driven excitable system. Physical Review Letters, 78, 775–778.

    Article  CAS  Google Scholar 

  • Pradines, J. R., Osipov, G. V., & Collins, J. J. (1999). Coherence resonance in excitable and oscillatory systems: The essential role of slow and fast dynamics. Physical Review E, 60, 6407–6410.

    Article  CAS  Google Scholar 

  • Rabinovich, M., Huerta, R., & Laurent, G. (2008). Transient dynamics for neural processing. Science, 321, 48–50.

    Article  CAS  PubMed  Google Scholar 

  • Rabinovich, M. I., & Varona, P. (2011). Robust transient dynamics and brain functions. Frontiers in Computational Neuroscience, 5, 24.

    Article  PubMed  PubMed Central  Google Scholar 

  • Rappel, W. J., & Karma, A. (1996). Noise-induced coherence in neural networks. Physical Review Letters, 77, 3251–3259.

    Article  Google Scholar 

  • Redish, A. D., Elga, A. N., & Touretzky, D. S. (1996). A coupled attractor model of the rodent head direction system. Network: Computation in Neural Systems, 7, 671–685.

    Article  Google Scholar 

  • Renart, A., & Machens, C. K. (2014). Variability in neural activity and behavior. Current Opinion in Neurobiology, 25, 211–220.

    Article  CAS  PubMed  Google Scholar 

  • Renart, A., & Machens, C. K. (2014). Variability in neural activity and behavior. Current Opinion in Neurobiology, 25, 211–220.

    Article  CAS  PubMed  Google Scholar 

  • Richardson, M. J. E. (2008). Spike-train spectra and network response functions for non-linear integrate-and-fire neurons. Biological Cybernetics, 99, 381–392.

    Article  PubMed  Google Scholar 

  • Richardson, M. J. E., Brunel, N., & Hakim, V. (2003). From subthreshold to firing-rate resonance. Journal of Neurophysiology, 89, 2538–2554.

    Article  PubMed  Google Scholar 

  • Risken, H. (1989). The Fokker-Planck equation (2nd ed.). Berlin: Springer-Verlag.

    Book  Google Scholar 

  • Robbe, L. T., Goris, J., Movshon, A., & EP, Smincelli. (2014). Partitioning neuronal variability. Nature Neuroscience, 17(6), 858–865.

    Article  CAS  Google Scholar 

  • Robinson, P. C., & Harsch, P. C. (2002). Stages of spike time variability during neuronal responses to transient inputs. Physical Review E, 66, 061902.

    Article  CAS  Google Scholar 

  • Romo, Ranulfo, Hernández, Adrián, Zainos, Antonio, & Salinas, Emilio. (2003). Correlated neuronal discharges that increase coding efficiency during perceptual discrimination. Neuron, 38(4), 649–657.

    Article  CAS  PubMed  Google Scholar 

  • Rotstein, H. G. (2014). Frequency preference response to oscillatory inputs in two-dimensional neural models: a geometric approach to subthreshold amplitude and phase resonance. The Journal of Mathematical Neuroscience, 4, 11.

    Article  PubMed  Google Scholar 

  • Rotstein, H. G. (2015). Subthreshold amplitude and phase resonance in models of quadratic type: nonlinear effects generated by the interplay of resonant and amplifying currents. Journal of Computational Neuroscience, 38, 325–354.

    Article  PubMed  Google Scholar 

  • Rotstein, H. G., & Nadim, F. (2014). Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents. Journal of Computational Neuroscience, 37, 9–28.

    Article  PubMed  Google Scholar 

  • Rotstein, H. G., Oppermann, T., White, J. A., & Kopell, N. (2006). The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells. Journal of Computational Neuroscience, 21, 271–292.

    Article  PubMed  Google Scholar 

  • Rowat, P. F., & Elson, R. C. (2004). State-dependent effects of Na channel noise on neuronal burst generation. Journal of Computational Neuroscience, 16, 87–112.

    Article  PubMed  Google Scholar 

  • Samsonovich, A., & McNaughton, B. L. (1997). Path integration and cognitive mapping in a continuous attractor neural network model. Journal of Neuroscience, 17, 5900–5920.

    Article  CAS  PubMed  Google Scholar 

  • Schneidman, E., Freedman, B., & Segev, I. (1998). Ion channel stochasticity may be critical in determining the reliability and precision of spike timing. Neural Computation, 10, 1679–1703.

    Article  CAS  PubMed  Google Scholar 

  • Shadlen, M. N., & Newsome, W. T. (1998). The variable discharge of cortical neurons: implications for connectivity, computation, and information coding. Journal of Neuroscience, 18, 3870–3896.

    Article  CAS  PubMed  Google Scholar 

  • Shalinsky, J. H., Magistretti, J., Ma, L., & Alonso, A. A. (2002). Muscarinic activation of a cation current and associated current noise in entorhinal-cortex layer-II neurons. Journal of Neurophysiology, 88, 1197–1211.

    Article  CAS  PubMed  Google Scholar 

  • Sigworth, F. J. (1980). The variance of sodium current fluctuations at the node of ranvier. Journal of Physiology (London), 307, 97–129.

    Article  CAS  Google Scholar 

  • Softky, W. R., & Koch, C. (1993). The highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps. Journal of Neuroscience, 13(1), 334–350.

    Article  CAS  PubMed  Google Scholar 

  • Stopfer, M., Bhagavan, S., Smith, B. H., & Laurent, G. (1997). Impared odor discrimination on desynchronization of odor-encoding neural assemblies. Nature, 390, 70–74.

    Article  CAS  PubMed  Google Scholar 

  • Strogatz, S. H. (1994). Nonlinear Dynamics and Chaos. Reading MA: Addison Wesley.

    Google Scholar 

  • Tateno, T., & Pakdaman, K. (2004). Random dynamics of the Morris-Lecar neural model. Chaos, 14, 511–530.

    Article  PubMed  Google Scholar 

  • Thomas, P. J., & Lindner, B. (2019). Phase descriptions of a multidimensional Ornstein-Uhlenbeck process. Physical Review E, 99, 062221.

    Article  CAS  PubMed  Google Scholar 

  • Uhlenbeck, G. E., & Ornstein, L. S. (1930). On the theory of brownian motion. Physical Review, 36, 823–841.

    Article  CAS  Google Scholar 

  • Van Kampen, N. G. (2011). Stochastic Processes in Physics and Chemistry. North-Holland Personal Library.

  • White, J., Rubinstein, J., & Kay, A. (2000). Channel noise in neurons. Trends in Neurosciences, 23, 131–137.

    Article  CAS  PubMed  Google Scholar 

  • White, J. A., & Haas, J. S. (2001). Intrinsic noise from voltage-gated ion channels: Effects on dynamics and reliability in intrinsically oscillatory neurons. In Handbook of Biological Physics, 4, 257–278.

    Article  Google Scholar 

  • White, J. A., Klink, R., Alonso, A., & Kay, A. R. (1998). Noise from voltage-gated ion channels may influence neuronal dynamics in the entorhinal cortex. Journal of Neurophysiology, 80, 262–269.

    Article  CAS  PubMed  Google Scholar 

  • White, J. A., Rubinstein, J. T., & Kay, A. R. (2000). Channel noise in neurons. Trends in Neurosciences, 23(3), 131–137.

    Article  CAS  PubMed  Google Scholar 

  • Wiesenfeld, K., & Moss, F. (1995). Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs. Nature, 373, 33–36.

    Article  CAS  PubMed  Google Scholar 

  • Yarom, Y., & Hounsgaard, J. (2011). Voltage fluctuations in neurons: Signal or noise? Physiological Reviews, 91, 917–929.

    Article  PubMed  Google Scholar 

  • Zhang, F., Wang, L.-P., Brauner, M., Lewwald, J. F., Kay, K., Watzke, N., et al. (2007). Multimodal fast optical interrogation of neural circuitry. Nature, 446, 633–641.

    Article  CAS  PubMed  Google Scholar 

  • Zhang, K. (1996). Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory. Journal of Neuroscience, 16, 2112–2126.

    Article  CAS  PubMed  Google Scholar 

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Acknowledgements

This work was partially supported by the National Science Foundation grant DMS-1608077 (HGR).

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Authors and Affiliations

  1. Federated Department of Biological Sciences, New Jersey Institute of Technology and Rutgers University, Newark, NJ, USA

    Rodrigo F. O. Pena & Horacio G. Rotstein

  2. Corresponding Investigator, CONICET, Buenos Aires, Argentina

    Horacio G. Rotstein

  3. Graduate Faculty, Behavioral Neurosciences Program, Rutgers University, Newark, NJ, USA

    Horacio G. Rotstein

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  1. Rodrigo F. O. Pena
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Appendices

Intrinsic and resonant oscillatory properties of 2D linear systems

Consider

$$\begin{aligned} \left\{ \begin{array}{ll} x' = a\, x + b\, y + A_{in}\, e^{i\, \omega \, t},\\ y' = c\, x+ d\, y \end{array} \right. \end{aligned}$$
(10)

where a, b, c and d are constants, \(\omega = 2 \pi f / 1000 > 0\) is the input frequency and \(A_{in} \ge 0\) is the input amplitude. The prime sign represents the derivative with respect to t. The units of t are ms and the units of f are Hz.

1.1 Intrinsic oscillations

The characteristic polynomial for the corresponding homogeneous system (\(A_{in} = 0\)) is given by

$$\begin{aligned} r^2 - (a + d)\, r+ ( a\, d - b\, c ) = 0. \end{aligned}$$
(11)

The eigenvalues are given by

$$\begin{aligned} r_{1,2} = \frac{a+d \pm \sqrt{(a-d)^2+4 b c}}{2}, \end{aligned}$$
(12)

and the natural (intrinsic) frequency of the (damped) oscillations (in Hz if t has units of ms) is given by

$$\begin{aligned} f_{nat} = \frac{\sqrt{-(a-d)^2-4 b c}}{4 \pi }\, 1000 \end{aligned}$$
(13)

assuming \((a-d)^2+4 b c < 0\).

1.2 Resonance and the impedance amplitude profile

The impedance amplitude profile \(Z(\omega )\) for system (10)-(11) is the magnitude

$$\begin{aligned} Z(\omega ) = \sqrt{\frac{d^2 + \omega ^2}{(a\, d - b\, c - \omega ^2)^2 + (a+d)^2\, \omega ^2}} \end{aligned}$$
(14)

of the complex valued coefficient of the particular solution to the system

$$\begin{aligned} \mathbf{Z}(\omega ) = \frac{(-d+i\, \omega )}{(-a\, + i\, \omega )\, (-d+i\, \omega ) - b\, c}. \end{aligned}$$
(15)

For 1D system, these quantities are given, respectively, by

$$\begin{aligned} Z(\omega ) = \frac{1}{\sqrt{a^2+\omega ^2}} \end{aligned}$$
(16)

and

$$\begin{aligned} \mathbf{Z}(\omega ) = \frac{1}{(-a\, + i\, \omega )}. \end{aligned}$$
(17)

The resonance frequency \(f_{res}\) (in Hz if t has units of ms) is the frequency at which Z reaches its maximum

$$\begin{aligned} f_{res} = \frac{\sqrt{ -d^2 + \sqrt{b^2\, c^2 - 2\, a\, b\, c\, d - 2\, d^2\, b\, c}}}{2 \pi }\, 1000. \end{aligned}$$
(18)

1.3 Response to constant inputs

The equilibrium solution to system (10) for a constant input \(A_{in}\) (i.e., \(\omega = 0\)) is given by

$$\begin{aligned} \bar{x} = -\frac{A_{in}\, d}{a d - b c} \quad \text{and} \quad \bar{y} = \frac{A_{in}\, c}{a d - b c}. \end{aligned}$$
(19)

The eigenvectors are given by

$$\begin{aligned} z_{1,2} = [b\ \ (r_{1,2}-a)]^T. \end{aligned}$$
(20)

The solution satisfying the initial conditions \([x(0)\ \ y(0)]^T = [x_0\ \ y_0]^T\) is given by

$$\begin{aligned} \left[ \begin{array}{lll} x \\ y \end{array} \right] = c_1\, \left[ \begin{array}{lll} \ \ \ \ \ b \\ r_1 -a \end{array} \right] \, e^{r_1 t} + c_2\, \left[ \begin{array}{lll} \ \ \ \ \ b \\ r_2 -a \end{array} \right] \, e^{r_2 t} + \left[ \begin{array}{lll} \bar{x} \\ \bar{y} \end{array} \right] \end{aligned}$$
(21)

where

$$\begin{aligned} &c_1 = \frac{b\, (y_0 - \bar{y}) - (x_0 - \bar{x})\, (r_2 - a)}{b\, (r_1 - r_2)} \quad \text{ and } \\& c_2 = \frac{-b\, (y_0 - \bar{y}) + (x_0 - \bar{x})\, (r_1 - a)}{b\, (r_1 - r_2)} \end{aligned}$$
(22)

For 1D systems (\(b = 0\)),

$$\begin{aligned} \bar{x} = -\frac{A_{in}}{a} \end{aligned}$$
(23)

and

$$\begin{aligned} x = \left( x_0 - \frac{A_{in}}{a} \right) e^{a t} -\frac{A_{in}}{a}, \end{aligned}$$
(24)

where \(x(0) = x_0\)

Ornstein-uhlenbeck (OU) process

1.1 One-dimensional OU process

The 1D OU process Uhlenbeck and Ornstein (1930) is described by the following linear stochastic differential equation

$$\begin{aligned} X' = -a X + I + \sigma \eta (t) \end{aligned}$$
(25)

where \(a>0\), I and \(\sigma\) are constants and \(\eta (t)\) is zero-mean and \(\delta\)-correlated Gaussian white noise. The parameter a is the inverse of the time constant and measure the strength by which the system reacts to perturbations. The parameter \(\sigma\) measures the intensity of the noise. The quotient \(I/\alpha\) is the asymptotic mean.

Using standard methods Risken (1989); Gardiner (1985) one can compute the solution satisfying \(X(0) = x_0\), which is the sum of a deterministic function with the form (24) and an integral of a deterministic function with respect to a Wiener process. The solution is normally distributed with mean and variance given, respectively by

$$\begin{aligned} E[X(t)] = \left( X_0 - \frac{I}{a} \right) e^{-a t} +\frac{I}{a} \end{aligned}$$
(26)

and

$$\begin{aligned} Var[X(t)] = \frac{\sigma ^2}{2 a}\, (1 - e^{-2 a t}). \end{aligned}$$
(27)

1.2 Higher-dimensional OU process

The multivariate OU process Uhlenbeck and Ornstein (1930) is described by the following linear stochastic differential equation

$$\begin{aligned} X' = A X + B + \Sigma H(t) \end{aligned}$$
(28)

where X is an n-dimensional vector, A is an \(n \times n\) matrix, B is an n-dimensional vector, \(\Sigma\) is an \(n \times m\) matrix and H is a vector of independent zero-mean and \(\delta\)-correlated Gaussian white noise components. Using standard methods Risken (1989); Gardiner (1985) one can compute the solution satisfying \(X(0) = x_0\). The solution is normally distributed. The mean is given by

$$\begin{aligned} E[X(t)] = [\, e^{A t} - B\, ]\, A^{-1} B + e^{A t} X(0), \end{aligned}$$
(29)

and the covariance matrix is given by

$$\begin{aligned} Cov[X(t)] = \int _0^t e^{A s} \Sigma \Sigma ^T e^{A^T s}. \end{aligned}$$
(30)

Under certain conditions, the covariance matrix corresponding to the stationary solutions reads

$$\begin{aligned} Cov[X(t)] = -\frac{\Sigma \Sigma ^T}{2\, Tr(A)} - \frac{[A - Tr(A) I]\, \Sigma \Sigma ^T\, [A - Tr(A) I]^T}{2\, Tr(A)\, det(A)}. \end{aligned}$$
(31)

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Pena, R.F.O., Rotstein, H.G. Oscillations and variability in neuronal systems: interplay of autonomous transient dynamics and fast deterministic fluctuations. J Comput Neurosci 50, 331–355 (2022). https://doi.org/10.1007/s10827-022-00819-7

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  • DOI: https://doi.org/10.1007/s10827-022-00819-7

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