It is widely appreciated that balanced excitation and inhibition are necessary for proper function in neural networks. However, in principle, balance could be achieved by many possible configurations of excitatory and inhibitory synaptic strengths and relative numbers of excitatory and inhibitory neurons. For instance, a given level of excitation could be balanced by either numerous inhibitory neurons with weak synapses or a few inhibitory neurons with strong synapses. Among the continuum of different but balanced configurations, why should any particular configuration be favored? Here, we address this question in the context of the entropy of network dynamics by studying an analytically tractable network of binary neurons. We find that entropy is highest at the boundary between excitation-dominant and inhibition-dominant regimes. Entropy also varies along this boundary with a trade-off between high and robust entropy: weak synapse strengths yield high network entropy which is fragile to parameter variations, while strong synapse strengths yield a lower, but more robust, network entropy. In the case where inhibitory and excitatory synapses are constrained to have similar strength, we find that a small, but non-zero fraction of inhibitory neurons, like that seen in mammalian cortex, results in robust and relatively high entropy.
REFERENCES
1.
M. D.
Fox
and M. E.
Raichle
, “Spontaneous fluctuations in brain activity observed with functional magnetic resonance imaging
,” Nat. Rev. Neurosci.
8
, 700
(2007
). 2.
A.
Arieli
, A.
Sterkin
, A.
Grinvald
, and A.
Aertsen
, “Dynamics of ongoing activity: Explanation of the large variability in evoked cortical responses
,” Science
273
, 1868
–1871
(1996
). 3.
A.
Luczak
, P.
Barthó
, and K. D.
Harris
, “Spontaneous events outline the realm of possible sensory responses in neocortical populations
,” Neuron
62
, 413
–425
(2009
). 4.
F.
Han
, N.
Caporale
, and Y.
Dan
, “Reverberation of recent visual experience in spontaneous cortical waves
,” Neuron
60
, 321
–327
(2008
). 5.
P.
Berkes
, G.
Orbán
, M.
Lengyel
, and J.
Fiser
, “Spontaneous cortical activity reveals hallmarks of an optimal internal model of the environment
,” Science
331
, 83
–87
(2011
). 6.
S. A.
Romano
, T.
Pietri
, V.
Pérez-Schuster
, A.
Jouary
, M.
Haudrechy
, and G.
Sumbre
, “Spontaneous neuronal network dynamics reveal circuit’s functional adaptations for behavior
,” Neuron
85
, 1070
–1085
(2015
). 7.
J.-e. K.
Miller
, I.
Ayzenshtat
, L.
Carrillo-Reid
, and R.
Yuste
, “Visual stimuli recruit intrinsically generated cortical ensembles
,” Proc. Natl. Acad. Sci.
111
, E4053
–E4061
(2014
). 8.
A. S.
Ecker
, P.
Berens
, R. J.
Cotton
, M.
Subramaniyan
, G. H.
Denfield
, C. R.
Cadwell
, S. M.
Smirnakis
, M.
Bethge
, and A. S.
Tolias
, “State dependence of noise correlations in macaque primary visual cortex
,” Neuron
82
, 235
–248
(2014
). 9.
D.
Lee
, N. L.
Port
, W.
Kruse
, and A. P.
Georgopoulos
, “Variability and correlated noise in the discharge of neurons in motor and parietal areas of the primate cortex
,” J. Neurosci.
18
, 1161
–1170
(1998
). 10.
B. B.
Averbeck
, P. E.
Latham
, and A.
Pouget
, “Neural correlations, population coding and computation
,” Nat. Rev. Neurosci.
7
, 358
(2006
). 11.
E. D.
Fagerholm
, G.
Scott
, W. L.
Shew
, C.
Song
, R.
Leech
, T.
Knöpfel
, and D. J.
Sharp
, “Cortical entropy, mutual information and scale-free dynamics in waking mice
,” Cereb. Cortex
26
, 3945
–3952
(2016
). 12.
W. L.
Shew
, H.
Yang
, S.
Yu
, R.
Roy
, and D.
Plenz
, “Information capacity and transmission are maximized in balanced cortical networks with neuronal avalanches
,” J. Neurosci.
31
, 55
–63
(2011
). 13.
B.-Q.
Mao
, F.
Hamzei-Sichani
, D.
Aronov
, R. C.
Froemke
, and R.
Yuste
, “Dynamics of spontaneous activity in neocortical slices
,” Neuron
32
, 883
–898
(2001
). 14.
S. H.
Gautam
, T. T.
Hoang
, K.
McClanahan
, S. K.
Grady
, and W. L.
Shew
, “Maximizing sensory dynamic range by tuning the cortical state to criticality
,” PLoS Comput. Biol.
11
, e1004576
(2015
). 15.
X.
Chen
and R.
Dzakpasu
, “Observed network dynamics from altering the balance between excitatory and inhibitory neurons in cultured networks
,” Phys. Rev. E
82
, 031907
(2010
). 16.
R. F.
Hunt
, K. M.
Girskis
, J. L.
Rubenstein
, A.
Alvarez-Buylla
, and S. C.
Baraban
, “Gaba progenitors grafted into the adult epileptic brain control seizures and abnormal behavior
,” Nat. Neurosci.
16
, 692
–697
(2013
). 17.
M. S. A.
Ferraz
, H. L. C.
Melo-Silva
, and A. H.
Kihara
, “Optimizing information processing in neuronal networks beyond critical states
,” PloS One
12
, e0184367
(2017
). 18.
S.
Scarpetta
and A.
de Candia
, “Neural avalanches at the critical point between replay and non-replay of spatiotemporal patterns
,” PLoS One
8
, e64162
(2013
). 19.
D.-P.
Yang
, H.-J.
Zhou
, and C.
Zhou
, “Co-emergence of multi-scale cortical activities of irregular firing, oscillations and avalanches achieves cost-efficient information capacity
,” PLoS Comput. Biol.
13
, e1005384
(2017
). 20.
S.
Denève
and C. K.
Machens
, “Efficient codes and balanced networks
,” Nat. Neurosci.
19
, 375
–382
(2016
). 21.
M.
Wehr
and A. M.
Zador
, “Balanced inhibition underlies tuning and sharpens spike timing in auditory cortex
,” Nature
426
, 442
–446
(2003
). 22.
B.
Haider
, A.
Duque
, A. R.
Hasenstaub
, and D. A.
McCormick
, “Neocortical network activity in vivo is generated through a dynamic balance of excitation and inhibition
,” J. Neurosci.
26
, 4535
–4545
(2006
). 23.
S.
Sahara
, Y.
Yanagawa
, D. D.
O’Leary
, and C. F.
Stevens
, “The fraction of cortical gabaergic neurons is constant from near the start of cortical neurogenesis to adulthood
,” J. Neurosci.
32
, 4755
–4761
(2012
). 24.
S.
Hendry
, H.
Schwark
, E.
Jones
, and J.
Yan
, “Numbers and proportions of gaba-immunoreactive neurons in different areas of monkey cerebral cortex
,” J. Neurosci.
7
, 1503
–1519
(1987
). 25.
D. L.
Meinecke
and A.
Peters
, “Gaba immunoreactive neurons in rat visual cortex
,” J. Comp. Neurol.
261
, 388
–404
(1987
). 26.
D. B.
Larremore
, W. L.
Shew
, E.
Ott
, F.
Sorrentino
, and J. G.
Restrepo
, “Inhibition causes ceaseless dynamics in networks of excitable nodes
,” Phys. Rev. Lett.
112
, 138103
(2014
). 27.
R. V.
Williams-García
, M.
Moore
, J. M.
Beggs
, and G.
Ortiz
, “Quasi-critical brain dynamics on a non-equilibrium widom line
,” Phys. Rev. E
90
, 062714
(2014
). 28.
S.
Song
, P. J.
Sjöström
, M.
Reigl
, S.
Nelson
, and D. B.
Chklovskii
, “Highly nonrandom features of synaptic connectivity in local cortical circuits
,” PLoS Biol.
3
, e68
(2005
). 29.
E.
Lehmann
and G.
Casella
, Theory of Point Estimation, Springer Texts in Statistics (Springer-Verlag, New York, 1998).30.
C.
van Vreeswijk
and H.
Sompolinsky
, “Chaotic balanced state in a model of cortical circuits
,” Neural Comput.
10
, 1321
–1371
(1998
). 31.
R.
Rubin
, L. F.
Abbott
, and H.
Sompolinsky
, “Balanced excitation and inhibition are required for high-capacity, noise-robust neuronal selectivity
,” Proc. Natl. Acad. Sci. U.S.A.
114
, E9366
–E9375
(2017
). 32.
S.
Denève
and C. K.
Machens
, “Efficient codes and balanced networks
,” Nat. Neurosci.
19
, 375
–382
(2016
). 33.
K. D.
Harris
and A.
Thiele
, “Cortical state and attention
,” Nat. Rev. Neurosci.
12
, 509
(2011
). 34.
M. A.
Dichter
and G.
Ayala
, “Cellular mechanisms of epilepsy: A status report
,” Science
237
, 157
–164
(1987
). 35.
F.
Fernandez
and C. C.
Garner
, “Over-inhibition: A model for developmental intellectual disability
,” Trends Neurosci.
30
, 497
–503
(2007
). 36.
J.
Rubenstein
and M. M.
Merzenich
, “Model of autism: Increased ratio of excitation/inhibition in key neural systems
,” Genes Brain Behav.
2
, 255
–267
(2003
). 37.
S. B.
Nelson
and V.
Valakh
, “Excitatory/inhibitory balance and circuit homeostasis in autism spectrum disorders
,” Neuron
87
, 684
–698
(2015
). 38.
N.
Gogolla
, J. J.
LeBlanc
, K. B.
Quast
, T. C.
Südhof
, M.
Fagiolini
, and T. K.
Hensch
, “Common circuit defect of excitatory-inhibitory balance in mouse models of autism
,” J. Neurodev. Disord.
1
, 172
(2009
). © 2018 Author(s).
2018
Author(s)
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