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**Bifurcations of large networks of two-dimensional integrate and fire neurons.**
*(English)*
Zbl 1276.92018

Summary: Recently, a class of two-dimensional integrate and fire models has been used to faithfully model spiking neurons. This class includes the Izhikevich model, the adaptive exponential integrate and fire model, and the quartic integrate and fire model. The bifurcation types for the individual neurons have been thoroughly analyzed by J. Touboul [SIAM J. Appl. Math. 68, No. 4, 1045–1079 (2008; Zbl 1149.34027)]. However, when the models are coupled together to form networks, the networks can display bifurcations that an uncoupled oscillator cannot. For example, the networks can transition from firing with a constant rate to burst firing. This paper introduces a technique to reduce a full network of this class of neurons to a mean field model, in the form of a system of switching ordinary differential equations. The reduction uses population density methods and a quasi-steady state approximation to arrive at the mean field system. Reduced models are derived for networks with different topologies and different model neurons with biologically derived parameters. The mean field equations are able to qualitatively and quantitatively describe the bifurcations that the full networks display. Extensions and higher order approximations are discussed.

### MSC:

92C20 | Neural biology |

37N25 | Dynamical systems in biology |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

34C23 | Bifurcation theory for ordinary differential equations |

### Keywords:

bifurcation theory; mean field; large networks; bursting; population density methods; integrate and fire### Citations:

Zbl 1149.34027
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\textit{W. Nicola} and \textit{S. A. Campbell}, J. Comput. Neurosci. 35, No. 1, 87--108 (2013; Zbl 1276.92018)

### References:

[1] | Abbott, LF; Vreeswijk, C, Asynchronous states in networks of pulse-coupled oscillators, Learning and Memory, 48, 1483-1490, (1993) |

[2] | Apfaltrer, F; Ly, C; Tranchina, D, Population density methods for stochastic neurons with realistic synaptic kinetics: firing rate dynamics and fast computational methods, Network: Computation in Neural Systems, 17, 373-418, (2006) |

[3] | Bernardo, M; Budd, C; Champneys, A; Kowalczyk, P; Nordmark, A; Tost, G; Piiroinen, P, Bifurcations in non-smooth dynamical systems, SIAM Review, 50, 629-701, (2008) · Zbl 1168.34006 |

[4] | Brette, R; Gerstner, W, Adaptive exponential integrate-and-fire model as an effective description of neuronal activity, Journal of Neurophysiology, 94, 3637-3642, (2005) |

[5] | Casti, A; Omurtag, A; Sornborger, A; Kaplan, E; Knight, BW; Victor, J; Sirovich, L, A population study of integrate-and-fire-or-burst neurons, Neural Computation, 14, 957-986, (2002) · Zbl 0995.92015 |

[6] | Destexhe, A; Mainen, Z; Sejnowski, T; Koch, C (ed.); Segev, I (ed.), Kinetic models of synaptic transmission, (1998), Cambridge, MA |

[7] | Dhooge, A; Govaerts, W; Kuznetsov, YA, Matcont: a MATLAB package for numerical bifurcation analysis of odes, ACM Transactions on Mathematical Software, 29, 141-164, (2003) · Zbl 1070.65574 |

[8] | Dur-e-Ahmad, M; Nicola, W; Campbell, SA; Skinner, F, Network bursting using experimentally constrained single compartment CA3 hippocampal neuron models with adaptation, Journal of Computational Neuroscience, 33, 21-40, (2012) |

[9] | Ermentrout, G.B., & Terman, D.H. (2010). Mathematical Foundations of Neuroscience. New York, NY: Springer. · Zbl 1320.92002 |

[10] | Fitzhugh, R, Impulses and phsyiological states in theoretical models of nerve membrane, Biophysical Journal, 1, 445-466, (1952) |

[11] | Gerstner, W., & Kistler, W. (2002). Spiking Neuron Models. Cambridge, UK: Cambridge University Press. · Zbl 1100.92501 |

[12] | Hines, ML; Morse, T; Migliore, M; Carnevale, NT; Shepherd, GM, Modeldb: a database to support computational neuroscience, Journal of Computational Neuroscience, 17, 7-11, (2004) |

[13] | Hemond, P; Epstein, D; Boley, A; Migliore, M; Ascoli, G; Jaffe, D, Distinct classes of pyramidal cells exhibit mutually exclusive firing patterns in hippocampal area CA3b, Hippocampus, 18, 411-424, (2008) |

[14] | Ho, EC; Zhang, L; Skinner, FK, No article title, Hippocampus, 19, 152-165, (2009) |

[15] | Izhikevich, E, Simple model of spiking neurons, Neural Networks, IEEE Transactions, 14, 1569-1572, (2003) |

[16] | Knight, BW, Dynamics of encoding in neuron populations: some general mathematical features, Neural Computation, 12, 473-518, (2000) |

[17] | Camera, G; Rauch, A; Luscher, HR; Senn, W; Fusi, S, Minimal models of adapted neuronal response to in-vivo like input currents, Neural Computation, 16, 2101-2124, (2004) · Zbl 1055.92011 |

[18] | Camera, G; Giugliano, M; Senn, W; Fusi, S, The response of cortical neurons to in vivo-like input current: theory and experiment, Biological Cybernetics, 99, 279-301, (2008) · Zbl 1154.92011 |

[19] | Ly, C; Tranchina, D, A critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling, Neural Computation, 19, 2032-2092, (2007) · Zbl 1131.92016 |

[20] | Markram, H; Toledo-Rodriguez, M; Wang, Y; Gupta, A; Silberberg, G; Wu, C, Interneurons of the neocortical inhibitory system, Nature Reviews: Neuroscience, 5, 793-807, (2004) |

[21] | MATLAB (2012). Version 7.10.0 (R2012a). The MathWorks Inc. Massachusetts: Natick. |

[22] | Naud, R; Marcille, N; Clopath, C; Gerstner, W, Firing patterns in the adaptive exponential integrate-and-fire model, Biological Cybernetics, 99, 335-347, (2008) · Zbl 1161.92012 |

[23] | Nesse, W; Borisyuk, A; Bressloff, P, Fluctuation-driven rhythmogenesis in an excitatory neuronal network with slow adaptation, Journal of Computational Neuroscience, 25, 317-333, (2008) |

[24] | Nykamp, D; Tranchina, D, A population density approach that facilitates large-scale modeling of neural networks: analysis and an application to orientation tuning, Journal of Computational Neuroscience, 8, 19-50, (2000) · Zbl 0999.92008 |

[25] | Omurtag, A; Knight, BW; Sirovich, L, On the simulation of large populations of neurons, Journal of Computational Neuroscience, 8, 51-63, (2000) · Zbl 1036.92010 |

[26] | Sirovich, L; Omurtag, A; Knight, BW, Dynamics of neuronal populations: the equilibrium solution, SIAM Journal on Applied Mathematics, 60, 2009-2028, (2000) · Zbl 0991.92005 |

[27] | Sirovich, L; Omurtag, A; Lubliner, K, Dynamics of neural populations: stability and synchrony, Network: Computation in Neural Systems, 17, 3-29, (2006) |

[28] | Strogatz, S; Mirollo, RE, Stability of incoherence in a population of coupled oscillators, Journal of Statistical Physics, 63, 613-635, (1991) |

[29] | Tikhonov, A, Systems of differential equations containing small parameters in the derivatives (in Russian), Matematicheskii Sbornik (NS), 31, 575-586, (1952) |

[30] | Touboul, J, Bifurcation analysis of a general class ofnonlinear integrate-and-fire neurons, SIAM Journal on Applied Mathematics, 68, 1045-1079, (2008) · Zbl 1149.34027 |

[31] | Treves, A, Mean-field analysis of neuronal spike dynamics, Network: Computation in Neural Systems, 4, 259-284, (1993) · Zbl 0798.92009 |

[32] | Vreeswijk, C, Partial synchronization in populations of pulse-coupled oscillators, Physical Review E, 54, 5522-5537, (1996) |

[33] | Vreeswijk, C; Hansel, D, Patterns of synchrony in neural networks with spike adaptation, Neural Computation, 13, 959-992, (2001) · Zbl 1004.92011 |

[34] | Vreeswijk, C; Abbott, LF; Ermentrout, GB, When inhibition not excitation synchronizes neural firing, Journal of Computational Neuroscience, 1, 313-321, (1994) |

[35] | Vladimirski, BB; Tabak, J; O’Donovan, MJ; Rinzel, J, Episodic activity in a heterogeneous excitatory network, from spiking neurons to Mean field, Journal of Computational Neuroscience, 25, 39-63, (2008) · Zbl 1412.92005 |

[36] | Wu, Y; Lu, W; Lin, W; Leng, G; Feng, J, Bifurcations of emergent bursting in a neuronal network, PLoS ONE, 7, e38402, (2012) |

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