Memristive biophysical neuron models forming an excitatory–inhibitory neural network for modeling PING rhythm generation

Abstract

SPICE models are constructed for memristive devices to form associated biophysical neuron circuit models such as the Hodgkin–Huxley (HH) type II excitability neuron circuit model, the HH type III excitability neuron circuit model, the simplified HH neuron circuit model, the Morris–Lecar neuron circuit model, and the memristive based direct-current (DC) circuit model. Rigorous nonlinear circuit-theoretic principles are also applied to analyze the different behaviors of the generic memristor Na\(^{+}\)-ion, K\(^{+}\)-ion, and Ca\(^{++}\)-ion channels forming these biophysical neuron circuit models. Detailed explanations and clarifications are presented on the memristive HH type II and HH type III axonal excitabilities based on mathematical analysis as well as the circuit models. This is done from the perspective of the spike patterns generated by both of these biophysical neuron circuit models. Moreover, various experimental studies have revealed a synchronous brain state known as gamma rhythms that are responsible for sensory, memory, and motor processes. This suggests that understanding how the gamma oscillation (30–100 Hz) is generated in the brain will be extremely important to unravel the link between the activity of an individual neuron and the cognitive processing achieved by a population of networked neurons. We thus also study the dynamics of an interconnected excitatory–inhibitory (E–I) network population, which is ubiquitous in the brain. Utilizing biophysical models of the E–I network, we investigate the generation of pyramidal-interneuronal network gamma (PING) rhythms caused by the external input to the network and the connectivity heterogeneities. The results reveal that synchronous strong PING and sparsely firing weak PING rhythms are generated based on the network connectivities and external input heterogeneities in simulations of 100 memristive HH type II excitability neurons forming an E–I network.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29

References

  1. 1.

    Chua, L.O.: Memristor, Hodgkin–Huxley, and edge of chaos. Nanotechnology (2013). https://doi.org/10.1088/0957-4484/24/38/383001

    Article  MATH  Google Scholar 

  2. 2.

    Sah, M. P., Kim, H., Chua, L. O.: Brains are made of memristors. In: IEEE Circuits and Systems Magazine (2014).https://doi.org/10.1109/MCAS.2013.2296414

  3. 3.

    Jo, S.H., Chang, T., Ebong, I., Bhadviya, B.B., Mazumder, P., Lu, W.: Nanoscale memristor device as synapse in neuromorphic systems. Nano Lett. (2010). https://doi.org/10.1021/nl904092h

    Article  Google Scholar 

  4. 4.

    Chua, L.O.: Memristor-The missing circuit element. In: IEEE Transactions on Circuit Theory(1971).https://doi.org/10.1109/TCT.1971.1083337

  5. 5.

    Strukov, D., Snider, G., Stewart, D., et al.: The missing memristor found. Nature (2008). https://doi.org/10.1038/nature06932

    Article  Google Scholar 

  6. 6.

    Waser, R.: Resistive non-volatile memory devices. Microelectron. Eng. (2009). https://doi.org/10.1016/j.mee.2009.03.132

    Article  Google Scholar 

  7. 7.

    Lehtonen, E., Poikonen, J.H., Laiho, M.: Two memristors suffice to compute all Boolean functions 1. Electron. Lett. (2010). https://doi.org/10.1049/el.2010.3407

    Article  Google Scholar 

  8. 8.

    Shin, S., Kim, K., Kang, S.: Memristor applications for programmable analog ICs. In: IEEE Transactions on Nanotechnology(2011) .https://doi.org/10.1109/TNANO.2009.2038610

  9. 9.

    Talukdar, A., Radwan, A.G., Salama, K.N.: Generalized model for memristor-based Wien-family oscillators. J. Microelectron. (2011). https://doi.org/10.1016/j.mejo.2011.07.001

    Article  Google Scholar 

  10. 10.

    Mosad, A.G., Fouda, M.E., Khatib, M.A., Salama, K.N., Radwan, A.G.: Improved memristor-based relaxation oscillator. Microelectron. J. (2013). https://doi.org/10.1016/j.mejo.2013.04.005

    Article  Google Scholar 

  11. 11.

    Radwan, A.G., Moaddy, K.: Shaher Momani: stability and non-standard finite difference method of the generalized Chua’s circuit. Comput. Math. Appl. (2011). https://doi.org/10.1016/j.camwa.2011.04.047

    Article  MATH  Google Scholar 

  12. 12.

    Chua, L.O., Sbitnev, V., Kim, H.: Hodgkin–Huxley axon is made of memristors. Int. J. Bifurc. Chaos (2012). https://doi.org/10.1142/S021812741230011X

    Article  MATH  Google Scholar 

  13. 13.

    Sah, M.P., Kim, H., Eroglu, A., Chua, L.O.: Memristive model of the Barnacle giant muscle fibers. Int. J. Bifurc. Chaos (2016). https://doi.org/10.1142/S0218127416300019

    Article  MATH  Google Scholar 

  14. 14.

    Nigus, M., Priyadarshini, R., Mehra, R.M.: Stochastic and novel generic scalable window function-based deterministic memristor SPICE model comparison and implementation for synaptic circuit design. SN Appl. Sci. (2020). https://doi.org/10.1007/s42452-019-1888-z

    Article  Google Scholar 

  15. 15.

    Herculano-Houzel, S.: The human brain in numbers: a linearly scaled-up primate brain. Front. Hum. Neurosci (2009). https://doi.org/10.3389/neuro.09.031.2009

    Article  Google Scholar 

  16. 16.

    Hodgkin, A., Huxley, A.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (1952). https://doi.org/10.1113/jphysiol.1952.sp004764

    Article  Google Scholar 

  17. 17.

    Hodgkin, A., Keynes, R.: Experiments on the injection of substances into squid giant axons by means of a microsyringe. J. Physiol. (1956). https://doi.org/10.1113/jphysiol.1956.sp005485

    Article  Google Scholar 

  18. 18.

    Young, J.Z.: Structure of nerve fibers and synapses in some invertebrates. Cold Spring Harbor Symp. Quant. Biol. (1936). https://doi.org/10.1101/SQB.1936.004.01.001

    Article  Google Scholar 

  19. 19.

    Izhikevich, E. M.: Simple model of spiking neurons. In: IEEE Transactions on Neural Networks (2003).https://doi.org/10.1109/TNN.2003.820440

  20. 20.

    Hindmarsh, J.L., Rose, R.M., Andrew, F.H.: A model of neuronal bursting using three coupled first order differential equations Proc. R. Soc. Lond. B. (1997). https://doi.org/10.1098/rspb.1984.0024

    Article  Google Scholar 

  21. 21.

    Ebong, I.E., Mazumder, P.: CMOS and memristor-based neural network design for position detection. Proc. IEEE (2012). https://doi.org/10.1109/JPROC.2011.2173089

    Article  Google Scholar 

  22. 22.

    Chua, L.O., Sbitnev, V., Kim, H.: Neurons are poised near the edge of chaos. Int. J. Bifurc. Chaos (2012). https://doi.org/10.1142/S0218127412500988

    Article  Google Scholar 

  23. 23.

    Chua, L.O.: Introduction to Nonlinear Network Theory. McGraw-Hill, New York (1969)

    Google Scholar 

  24. 24.

    Chua, L.O., Desoer, C.A., Kuh, E.S.: Linear and Nonlinear Circuits. McGraw-Hill, New York (1987). https://doi.org/10.1002/bimj.4710300726

    Google Scholar 

  25. 25.

    Cole, K.S.: Membranes, Ions and Impulses. University of California Press, Berkeley (1972)

    Google Scholar 

  26. 26.

    Hegab, A.M., Salem, N.M., Radwan, A.G., Chua, L.O.: Neuron model with simplified memristive ionic channels. Int. J. Bifurc. Chaos (2015). https://doi.org/10.1142/S0218127415300177

    Article  MATH  Google Scholar 

  27. 27.

    Hagiwara, S., Naka, K.I.: The initiation of spike potential in Barnacle muscle fibers under low intracellular Ca++. J. Gen. Physiol. (1964). https://doi.org/10.1085/jgp.48.1.141

    Article  Google Scholar 

  28. 28.

    Hagiwara, S.: Membrane properties of the Barnacle muscle fiber. Ann. NY Acad. Sci. (1966). https://doi.org/10.1111/j.1749-6632.1966.tb50213.x

    Article  Google Scholar 

  29. 29.

    Hagiwara, S., Hayashi, H., Takahashi, K.: Calcium and potassium currents of the membrane of a Barnacle muscle fiber in relation to the calcium spike. J. Physiol. (1969). https://doi.org/10.1113/jphysiol.1969.sp008955

    Article  Google Scholar 

  30. 30.

    Keynes, R.D., Rojas, E., Taylor, R.E., Vergara, J.: Calcium and potassium systems of a giant Barnacle muscle fiber under membrane potential control. J. Physiol. (1973). https://doi.org/10.1113/jphysiol.1973.sp010146

    Article  Google Scholar 

  31. 31.

    Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. (1981). https://doi.org/10.1016/S0006-3495(81)84782-0

    Article  Google Scholar 

  32. 32.

    Hodgkin, A.: The local electric changes associated with repetitive action in a non-medullated axon. J. Physiol. (1948). https://doi.org/10.1113/jphysiol.1948.sp004260

    Article  Google Scholar 

  33. 33.

    Zhao, Z., Li, L., Gu, H.: Different dynamical behaviors induced by slow excitatory feedback for type II and III excitabilities. Sci. Rep. (2020). https://doi.org/10.1038/s41598-020-60627-w

    Article  Google Scholar 

  34. 34.

    Clay, J.R.: Excitability of the squid giant axon revisited. J. Neurophysiol. (1998). https://doi.org/10.1152/jn.1998.80.2.903

    Article  Google Scholar 

  35. 35.

    Guttman, R., Lewis, S., Rinzel, J.: Control of repetitive firing in squid axon membrane as a model for a neuroneoscillator. J. Physiol. (1980). https://doi.org/10.1113/jphysiol.1980.sp013370

    Article  Google Scholar 

  36. 36.

    Clay, J.R., Paydarfar, D., Forger, D.B.: A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons. J. R. Soc. Interface (2008). https://doi.org/10.1098/rsif.2008.0166

    Article  Google Scholar 

  37. 37.

    Singer, W.: Neuronal synchrony: a versatile code for the definition of relations? Neuron (1999). https://doi.org/10.1016/s0896-6273(00)80821-1

    Article  Google Scholar 

  38. 38.

    Singer, W., Gray, C.M.: Visual feature integration and the temporal correlation hypothesis. Annu. Rev. Neurosci. (1995). https://doi.org/10.1146/annurev.ne.18.030195.003011

    Article  Google Scholar 

  39. 39.

    Sanes, J.N., Donoghue, J.P.: Oscillations in local field potentials of the primate motor cortex during voluntary movement. Proc. Natl. Acad. Sci. USA (1993). https://doi.org/10.1073/pnas.90.10.4470

    Article  Google Scholar 

  40. 40.

    Jensen, O., Kaiser, J., Lachaux, J.P.: Human gamma-frequency oscillations associated with attention and memory. Trends Neurosci. (2007). https://doi.org/10.1016/j.tins.2007.05.001

    Article  Google Scholar 

  41. 41.

    Bragin, A., Jandó, G., Nádasdy, Z., Hetke, J., Wise, K., Buzsáki, G.: Gamma (40–100 Hz) oscillation in the hippocampus of the behaving rat. J. Neurosci. (1995). https://doi.org/10.1523/JNEUROSCI.15-01-00047.1995

    Article  Google Scholar 

  42. 42.

    Yinghang, H., Yubing, G., Li, W., Xiaoguang, M., Chuanlu, Y.: Single or multiple synchronization transitions in scale-free neuronal networks with electrical or chemical coupling. Chaos Solitons Fractals (2011). https://doi.org/10.1016/j.chaos.2011.02.005

    MathSciNet  Article  Google Scholar 

  43. 43.

    Han, Fang, Wang, Zhijie, Ying, Du, Sun, Xiaojuan, Zhang, Bin: Robust synchronization of bursting Hodgkin–Huxley neuronal systems coupled by delayed chemical synapses. Int. J. Non-Linear Mech. (2015). https://doi.org/10.1016/j.ijnonlinmec.2014.10.010

    Article  Google Scholar 

  44. 44.

    Hájos, N., Paulsen, O.: Network mechanisms of gamma oscillations in the CA3 region of the hippocampus. Neural Netw. (2009). https://doi.org/10.1016/j.neunet.2009.07.024

    Article  Google Scholar 

  45. 45.

    Batista, C.A.S., Viana, R.L., Lopes, S.R., Batista, A.M.: Dynamic range in small-world networks of Hodgkin-Huxley neurons with chemical synapses. Phys. A Stat. Mech. Appl. (2014). https://doi.org/10.1016/j.physa.2014.05.069

    Article  MATH  Google Scholar 

  46. 46.

    Scott, R., Michal, Z., Victoria, B.: Effects of neuromodulation on excitatory–inhibitory neural network dynamics depend on network connectivity structure. J. Nonlinear Sci. (2014). https://doi.org/10.1007/s00332-017-9438-6

    Article  Google Scholar 

  47. 47.

    Kilpatrick, Z.P., Ermentrout, B.: Sparse gamma rhythms arising through clustering in adapting neuronal networks. PLoS Comput. Biol. (2011). https://doi.org/10.1371/journal.pcbi.1002281

    MathSciNet  Article  Google Scholar 

  48. 48.

    Hansel, D., Mato, G., Meunier, C.: Synchrony in excitatory neural networks. Neural Comput. (1995). https://doi.org/10.1162/neco.1995.7.2.307

    Article  Google Scholar 

  49. 49.

    Wang, X.J., Buzsáki, G.: Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model. J. Neurosci. (1996). https://doi.org/10.1523/JNEUROSCI.16-20-06402.1996

    Article  Google Scholar 

  50. 50.

    Traub, R.D., Whittington, M.A., Colling, S.B., Buzsáki, G., Jefferys, J.G.: Analysis of gamma rhythms in the rat hippocampus in vitro and in vivo. J. Physiol. (1996). https://doi.org/10.1113/jphysiol.1996.sp021397

    Article  Google Scholar 

  51. 51.

    Whittington, M.A., Traub, R.D., Kopell, N., Ermentrout, B., Buhl, E.H.: Inhibition-based rhythms: experimental and mathematical observations on network dynamics. Int. J. Psychophysiol. (2000). https://doi.org/10.1016/s0167-8760(00)00173-2

    Article  Google Scholar 

  52. 52.

    Brunel, N.: Dynamics of Sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci. (2000). https://doi.org/10.1023/A:1008925309027

    Article  MATH  Google Scholar 

  53. 53.

    Hansel, D., Mato, G.: Asynchronous states and the emergence of synchrony in large networks of interacting excitatory and inhibitory neurons. Neural Comput. (2003). https://doi.org/10.1162/089976603321043685

    Article  MATH  Google Scholar 

  54. 54.

    Börgers, C., Kopell, N.: Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity. Neural Comput. (2003). https://doi.org/10.1162/089976603321192059

    Article  MATH  Google Scholar 

  55. 55.

    Goldman, D.E.: Potential, impedance, and rectification in membranes. J. Gen. Physiol. (1943). https://doi.org/10.1085/jgp.27.1.37

    Article  Google Scholar 

  56. 56.

    Hodgkin, A., KATZ, B.: The effect of sodium ions on the electrical activity of giant axon of the squid. J. Physiol. (1949). https://doi.org/10.1113/jphysiol.1949.sp004310

    Article  Google Scholar 

  57. 57.

    Frankenhaeuser, B.: Potassium permeability in myelinated nerve fibres of Xenopus laevis. J. Physiol. (1962). https://doi.org/10.1113/jphysiol.1962.sp006834

    Article  Google Scholar 

  58. 58.

    Binstock, L., Goldman, L.: Rectification in instantaneous potassium current-voltage relations in Myxicola giant axons. J. Physiol. (1971). https://doi.org/10.1113/jphysiol.1971.sp009583

    Article  Google Scholar 

  59. 59.

    Clay, J.R.: A paradox concerning ion permeation of the delayed rectifier potassium ion channel in squid giant axons. J. Physiol. (1991). https://doi.org/10.1113/jphysiol.1991.sp018890

    Article  Google Scholar 

  60. 60.

    Clay, J.R., Shlesinger, M.F.: Effects of external cesium and rubidium on outward potassium currents in squid axons. Biophys. J. (1983). https://doi.org/10.1016/S0006-3495(83)84367-7

    Article  Google Scholar 

  61. 61.

    Clay, J.R.: Axonal excitability revisited. Prog. Biophys. Mol. Biol. (2005). https://doi.org/10.1016/j.pbiomolbio.2003.12.004

    Article  Google Scholar 

  62. 62.

    Chua, L. O., Sung M. K.: Memristive devices and systems. In: Proceedings of the IEEE(1976) .https://doi.org/10.1109/PROC.1976.10092

  63. 63.

    Chua, L.O.: Resistance switching memories are memristors. Appl. Phys. A (2011). https://doi.org/10.1007/s00339-011-6264-9

    Article  MATH  Google Scholar 

  64. 64.

    Chua, L.O.: If it’s pinched it’s a memristor. Semicond. Sci. Technol. (2014). https://doi.org/10.1088/0268-1242/29/10/104001

    Article  Google Scholar 

  65. 65.

    Traub, R.D.: Simulation of intrinsic bursting in CA3 hippocampal neurons. Neuroscience (1982). https://doi.org/10.1016/0306-4522(82)91130-7

    Article  Google Scholar 

Download references

Funding

Not applicable.

Author information

Affiliations

Authors

Contributions

M.N.G.: conceptualization, methodology, formal analysis and investigation, software, writing of original draft. R.P.: writing—review and editing, validation, supervision. R.M.M.: supervision.

Corresponding author

Correspondence to Rashmi Priyadarshini.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author (Dr. Rashmi Priyadarshini) states that there is no conflict of interest.

Code availability

Code will be made available upon request.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Getachew, M.N., Priyadarshini, R. & Mehra, R.M. Memristive biophysical neuron models forming an excitatory–inhibitory neural network for modeling PING rhythm generation. J Comput Electron (2020). https://doi.org/10.1007/s10825-020-01580-9

Download citation

Keywords

  • Memristive neuron models
  • Memristive neuron DC models
  • Memristor ion channels
  • HH type II excitability
  • HH type III excitability
  • PING rhythms