Symmetries constrain dynamics in a family of balanced neural networks. (English) Zbl 1395.92029

Summary: We examine a family of random firing-rate neural networks in which we enforce the neurobiological constraint of Dale’s Law – each neuron makes either excitatory or inhibitory connections onto its post-synaptic targets. We find that this constrained system may be described as a perturbation from a system with nontrivial symmetries. We analyze the symmetric system using the tools of equivariant bifurcation theory and demonstrate that the symmetry-implied structures remain evident in the perturbed system. In comparison, spectral characteristics of the network coupling matrix are relatively uninformative about the behavior of the constrained system.


92C20 Neural biology
92B20 Neural networks for/in biological studies, artificial life and related topics
37N25 Dynamical systems in biology
37G40 Dynamical aspects of symmetries, equivariant bifurcation theory


Full Text: DOI arXiv


[1] Watts, DJ; Strogatz, SH, Collective dynamics of ‘small-world’ networks, Nature, 393, 440-442, (1998) · Zbl 1368.05139
[2] Park, H-J; Friston, K, Structural and functional brain networks: from connections to cognition, Science, (2013)
[3] Hu Y, Brunton SL, Cain N, Mihalas S, Kutz JN, Shea-Brown E. Feedback through graph motifs relates structure and function in complex networks. arXiv:1605.09073 2016.
[4] Tao, T; Vu, V; Krishnapur, M, Random matrices: universality of ESDs and the circular law, Ann Probab, 38, 2023-2065, (2010) · Zbl 1203.15025
[5] Sompolinsky, H; Crisanti, A; Sommers, HJ, Chaos in random neural networks, Phys Rev Lett, 61, 259-262, (1988)
[6] Girko, V, Circular law, Theory Probab Appl, 29, 694-706, (1985)
[7] Sommers, HJ; Crisanti, A; Sompolinsky, H; Stein, Y, Spectrum of large random asymmetric matrices, Phys Rev Lett, 60, 1895-1898, (1988)
[8] Bai, ZD, Circular law, Ann Probab, 25, 494-529, (1997) · Zbl 0871.62018
[9] Vreeswijk, C; Sompolinsky, H, Chaos in neuronal networks with balanced excitatory and inhibitory activity, Science, 274, 1724-1726, (1996)
[10] Renart, A; Rocha, J; Bartho, P; Hollender, L; Parga, N; Reyes, A; Harris, KD, The asynchronous state in cortical circuits, Science, 327, 587-590, (2010)
[11] Rajan, K; Abbott, LF, Eigenvalue spectra of random matrices for neural networks, Phys Rev Lett, 97, (2006)
[12] Wei, Y, Eigenvalue spectra of asymmetric random matrices for multicomponent neural networks, Phys Rev E, 85, (2012)
[13] Tao, T, Outliers in the spectrum of iid matrices with bounded rank perturbations, Probab Theory Relat Fields, 155, 231-263, (2013) · Zbl 1261.60009
[14] Muir, DR; Mrsic-Flogel, T, Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks, Phys Rev E, 91, (2015)
[15] Ahmadian, Y; Fumarola, F; Miller, KD, Properties of networks with partially structured and partially random connectivity, Phys Rev E, 91, (2015)
[16] Aljadeff, J; Stern, M; Sharpee, T, Transition to chaos in random networks with cell-type-specific connectivity, Phys Rev Lett, 114, (2015)
[17] Hermann, G; Touboul, J, Heterogeneous connections induce oscillations in large-scale networks, Phys Rev Lett, 109, (2012)
[18] Cabana, T; Touboul, J, Large deviations, dynamics and phase transitions in large stochastic and disordered neural networks, J Stat Phys, 153, 211-269, (2013) · Zbl 1291.82099
[19] Kadmon, J; Sompolinsky, H, Transition to chaos in random neuronal networks, Phys Rev X, 5, (2015)
[20] Garcia del Molino, LC; Pakdaman, K; Touboul, J; Wainrib, G, Synchronization in random balanced networks, Phys Rev E, 88, (2013)
[21] Sussillo, D; Abbott, LF, Generating coherent patterns of activity from chaotic neural networks, Neuron, 63, 544-557, (2009)
[22] Rajan, K; Abbott, LF; Sompolinsky, H, Stimulus-dependent suppression of chaos in recurrent neural networks, Phys Rev E, 82, (2010)
[23] Ostojic, S, Two types of asynchronous activity in networks of excitatory and inhibitory spiking neurons, Nat Neurosci, 17, 594-600, (2014)
[24] Connelly, WM; Lees, G, Modulation and function of the autaptic connections of layer V fast spiking interneurons in the rat neocortex, J Physiol, 588, 2047-2063, (2010)
[25] Cicogna, G, Symmetry breakdown from bifurcation, Lett Nuovo Cimento (2), 31, 600-602, (1981)
[26] Golubitsky M, Stewart I, Schaeffer DG. Singularities and groups in bifurcation theory. Vol. II. New York: Springer; 1988. · Zbl 0691.58003
[27] Hoyle RB. Pattern formation: an introduction to methods. Cambridge: Cambridge University Press; 2006. (Cambridge texts in applied mathematics). · Zbl 1087.00001
[28] Dhooge, A; Govaerts, W; Kuznetsov, YA, MATCONT: a MATLAB package for numerical bifurcation analysis of odes, ACM Trans Math Softw, 29, 141-164, (2003) · Zbl 1070.65574
[29] Lauterbach R, Matthews P. Do absolutely irreducible group actions have odd dimensional fixed point spaces? arXiv:1011.3986v1 2010.
[30] Lauterbach, R; Schwenker, S, Equivariant bifurcations in 4-dimensional fixed point spaces, Dyn Syst, 32, 117-147, (2017) · Zbl 1362.37103
[31] Wilson, HR; Cowan, JD, Excitatory and inhibitory interactions in localized populations of model neurons, Biophys J, 12, 1-23, (1972)
[32] Wilson, HR; Cowan, JD, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13, 55-80, (1973) · Zbl 0281.92003
[33] Ermentrout GB, Terman D. Foundations of mathematical neuroscience. Berlin: Springer; 2010. · Zbl 1320.92002
[34] Bressloff, P, Spatiotemporal dynamics of continuum neural fields, J Phys A, Math Theor, 45, (2012) · Zbl 1263.92008
[35] Ginzburg, I; Sompolinsky, H, Theory of correlations in stochastic neural networks, Phys Rev E, 50, 3171-3191, (1994)
[36] Brunel, N; Hakim, V, Fast global oscillations in networks of integrate-and-fire neurons with low firing rates, Neural Comput, 11, 1621-1671, (1999)
[37] Ermentrout, GB; Cowan, JD, A mathematical theory of visual hallucination patterns, Biol Cybern, 34, 137-150, (1979) · Zbl 0409.92008
[38] Diekman, CO; Golubitsky, M, Network symmetry and binocular rivalry experiments, J Math Neurosci, 4, (2014) · Zbl 1321.92044
[39] Butera, RJ; Rinzel, J; Smith, JC, Models of respiratory rhythm generation in the pre-Bötzinger complex. I. bursting pacemaker neurons, J Neurophysiol, 82, 382-397, (1999)
[40] Marder, E; Bucher, D, Central pattern generators and the control of rhythmic movements, Curr Biol, 11, 986-996, (2001)
[41] Pearson, K, Common principles of motor control in vertebrates and invertebrates, Annu Rev Neurosci, 16, 265-297, (1993)
[42] Golubitsky, M; Stewart, I; Buono, P-L; Collins, JJ, A modular network for legged locomotion, Physica D, 115, 56-72, (1998) · Zbl 1039.92009
[43] Buono, P-L; Golubitsky, M, Models of central pattern generators for quadruped locomotion. I. primary gaits, J Math Biol, 42, 291-326, (2001) · Zbl 1039.92007
[44] Golubitsky, M; Shiau, LJ; Stewart, I, Spatiotemporal symmetries in the disynaptic canal-neck projection, SIAM J Appl Math, 67, 1396-1417, (2007) · Zbl 1129.92015
[45] Kriener, B; Helias, M; Rotter, S; Diesmann, M; Einevoll, GT, How pattern formation in ring networks of excitatory and inhibitory spiking neurons depends on the input current regime, Front Comput Neurosci, 7, (2014)
[46] Sussillo, D, Neural circuits as computational dynamical systems, Curr Opin Neurobiol, 25, 156-163, (2014)
[47] Barreiro, AK; Kutz, JN; Shlizerman, E, Symmetries constrain the transition to heterogeneous chaos in balanced networks, BMC Neurosci, 16, (2015)
[48] Litwin-Kumar, A; Doiron, B, Slow dynamics and high variability in balanced cortical networks with clustered connections, Nat Neurosci, 15, 1498-1505, (2012)
[49] Laurent, G, Olfactory network dynamics and the coding of multidimensional signals, Nat Rev Neurosci, 3, 884-895, (2002)
[50] Broome, BM; Jayaraman, V; Laurent, G, Encoding and decoding of overlapping odor sequences, Neuron, 51, 467-482, (2006)
[51] Yu, BM; Cunningham, JP; Santhanam, G; Ryu, SI; Shenoy, KV; Sahani, M, Gaussian-process factor analysis for low-dimensional single-trial analysis of neural population activity, J Neurophysiol, 102, 614-635, (2009)
[52] Machens, CK; Romo, R; Brody, CD, Functional, but not anatomical, separation of “what“ and “when” in prefrontal cortex, J Neurosci, 30, 350-360, (2010)
[53] Churchland, MM; Cunningham, JP; Kaufman, MT; Foster, JD; Nuyujukian, P; Ryu, SI; Shenoy, KV, Neural population dynamics during reaching, Nature, 487, 51-56, (2012)
[54] Shlizerman, E; Schroder, K; Kutz, J, Neural activity measures and their dynamics, SIAM J Appl Math, 72, 1260-1291, (2012) · Zbl 1318.92008
[55] Shlizerman, E; Riffell, J; Kutz, J, Data-driven inference of network connectivity for modeling the dynamics of neural codes in the insect antennal lobe, Front Comput Neurosci, 8, (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.