Tuning network dynamics from criticality to the chaotic balanced state

According to many experimental observations, neurons in cerebral cortex tend to operate in an asynchronous regime, firing independently of each other. In contrast, many other experimental observations reveal cortical population firing dynamics that are relatively coordinated and occasionally synchronous. These discrepant observations have naturally led to a lively debate surrounding discrepant hypotheses. A commonly hypothesized explanation of asynchronous firing is that excitatory and inhibitory neurons are precisely correlated, nearly canceling each other, resulting in the so-called ‘chaotic balanced state’. On the other hand, the ‘criticality’ hypothesis posits an explanation of the more coordinated state that also requires a certain balance of excitatory and inhibitory interactions. Both hypotheses claim the same qualitative mechanism - properly balanced excitation and inhibition. Thus, a natural question arises: how are the chaotic balanced state and criticality related, how do they differ? Here we propose an answer to this question based on investigation of a simple, network-level computational model. We show that the strength of inhibitory synapses relative to excitatory synapses can be tuned from weak to strong to generate a family of models that spans a continuum from ‘criticality’ to the ‘chaotic balanced state’. Our results bridge two long-standing competing hypotheses and offer a possible explanation of discrepant experimental observations: neuromodulatory mechanisms that tune the strength of excitatory and inhibitory synapses may tune cortical state from criticality to the balanced state, and to intermediate states between these extremes. Author summary What is the dynamical state of cerebral cortex? Are neurons mostly uncorrelated, firing asynchronously with each other? Are synchronous oscillations important? The answers to these questions have fundamental implications for how the cortical neural population encodes and processes information. Here we show that two possible scenarios - criticality and the chaotic balanced state - that are typically considered incompatible can be attained in the same network by maintaining a certain kind of balance, while tuning the strength of inhibition relative to excitation.

Mounting experimental evidence supports the hypothesis that the cerebral cortex 2 operates in a dynamical regime near criticality [1][2][3][4][5][6][7][8]. What do we mean by criticality? 3 Often, criticality is described as a boundary in the space of possible dynamical regimes. 4 On one side of the boundary population activity tends to be orderly, correlated across 5 the entire network. On the other side, neurons fire more independently of each other. 6 At criticality, population dynamics are more diverse, rarely exhibiting synchronization 7 that spans the network, but often showing coordinated firing among groups of neurons 8 at small and intermediate scales [9,10]. Direct evidence that the cerebral cortex may 9 indeed operate near such a boundary comes from experiments and models in which the 10 balance of excitation (E) and inhibition (I) is disrupted. These studies show that one 11 can push cortical dynamics from a dynamical regime consistent with criticality to a 12 hyperactive synchronous regime by suppressing inhibiton (GABA antagonists) or to a 13 low-firing asynchronous state by increasing inhibition (GABA agonists) [11][12][13][14][15]. 14 Similarly, critical dynamics can be pushed to a low-firing asynchronous regime by 15 suppressing excitation (AMPA and NMDA antagonists) [13][14][15]. These observations 16 support the hypothesis that the cortex may operate near criticality under normal 17 conditions, but only if the proper balance of E and I is maintained. 18 However, not all observations of the cortex under 'normal conditions' exhibit the 19 diverse multi-scale coordination that is expected near criticality. Indeed, many 20 experimental measurements have revealed relatively asynchronous firing, particularly in 21 vigilant and active behavioral conditions [16][17][18][19][20]. One of the most prominent theoretical 22 explanations of the asynchronous state, the so-called 'chaotic balanced state' hypothesis, 23 is based on balanced E and I [21][22][23][24]. The idea is that E and I inputs to any given 24 neuron wax and wane together, nearly canceling each other most of the time. Some 25 experiments support this possibility based on whole cell recordings of E and I 26 inputs [25][26][27]. During brief moments the E-I cancellation is imperfect and neurons can 27 fire. 28 Both the criticality hypothesis and the chaotic balanced state hypothesis seem to 29 require balanced E and I. Support for both hypotheses has been shown in awake 30 animals. However, the difference in coordination of population activity for criticality 31 versus the chaotic balanced state is stark. How can we reconcile these two hypotheses? 32 When should we expect to see the coordination of criticality; when should we expect to 33 see the asychronous activity of the chaotic balanced state? 34 Here we hypothesize that criticality requires a different kind of E/I balance than the 35 chaotic balanced state. Here we address this possibility using a network-level model of 36 probabilistic, binary neurons. By tuning a single parameter, we found that we can 37 generate a family of models, spanning a continuum from criticality to the chaotic 38 balanced state. When synapses are strong and balanced, the chaotic balanced state 39 results. When synapses are relatively weak and balanced, criticality results. Our results 40 offer a possible explanation for the variety of experimental observations, suggesting that 41 the cortex could shift its dynamical regime from near criticality to the chaotic balanced 42 state and a continuum of intermediate states between these extremes, all while 43 maintaining balanced excitation and inhibition.

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We study a recurrent network of N = 1000 probabilistic integrate-and-fire of binary 46 neurons. There are 800 excitatory neurons and 200 inhibitory neurons. By altering 47 excitatory and inhibitory interactions, we consider a range of different dynamical 48 regimes. We tune the relative strengths of excitatory and inhibitory synapses including 49 a range of I/E weight ratios between 0 and 4.6. For every I/E ratio, the synaptic weight 50 matrix is normalized such that its largest eigenvalue is unity in magnitude. This 51 constraint ensures that activity does not tend to grow nor decay on average as time 52 passes [28,29]. In this sense, the activity is stable for all the dynamical regimes we 53 consider, but the stability is achieved by different combinations of E and I. In some 54 previous work on criticality, the constraint of unity largest eigenvalue has been used to 55 define 'balance' of excitation and inhibiton, but this should not be confused with the 56 different meaning of 'balance' considered in previous studies of the chaotic balanced 57 state.

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Typical population dynamics of the system under different I/E ratios are shown in 59 Fig 1a. When the I/E weight ratio is low, population activity is rather synchronized.

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The population tends to fire in bursts of coordinated activity with diverse burst sizes, 61 separated by periods of quiescence. As the I/E weight ratio increases to moderate levels 62 (I/E≈2), the network becomes less synchronized, and counter-intuitively, the population 63 firing rate increases as the inhibition becomes stronger. However, this is consistent with 64 previous work showing that inhibitory neurons in a recurrent network can make the 65 dynamics ceaseless, provided that the largest eigenvalue of the weight matrix is 66 unity [30]. As we increase the I/E weight ratio further, the network is further 67 desynchronized and the population firing rate eventually decreases. As I/E is tuned 68 from 0 to 4.6, synapse strength increases due to imposing the constraint that the largest 69 eigenvalue of the synaptic weight matrix is unity (Fig 1b). Thus, the synchronized . Population-level coordination decreases dramatically, reaching an asynchronous regime as inhibition is strengthened relative to excitation. (b) Average excitatory (green) and inhibitory (red) synaptic weight as a function of the I/E weight ratio for each realization. Synapse strength increases as I/E ratio increases due to the constraint of unity largest eigenvalue.
To more carefully examine the dynamics of excitation and inhibition, we consider 73 separately the excitatory and inhibitory inputs to the model cells. For all I/E ratios, E 74 and I are correlated, but as we increase the I/E ratio, the dynamics are tuned from a 75 state in which excitatory input dominates (is not canceled by inhibition) to a state in 76 which inhibitory input cancels the excitatory input more and more exactly (Fig 2a). We 77 define 'E/I tension' to measure how tightly the excitatory and inhibitory inputs balance 78 each other (Methods). Fig 2b shows that the E/I tension gradually increases as the I/E 79 weight ratio increases, and reaches as high as 1 when the I/E weight ratio is 4. This 80 result is consistent with previous work showing that tightly balanced excitation and 81 inhibition that cancel each other leads to desychronization [23]. One way to test the criticality hypothesis is to examine distributions of neural 83 avalanche sizes [11]. Similar to previous work [11,30,31], we define an avalanche as a 84 period of time during which the number of active neurons exceed a threshold (Fig 3a). 85 The duration and size of an avalanche are defined as the number of time steps and the 86 total number of spikes that occurred during the avalanche, respectively. We found that 87 both avalanche duration and size distributions were sensitive to changing I/E (Fig 3b). 88 When the I/E weight ratio is low, the avalanche duration and size distributions are close 89 to power-law distributions, as expected at criticality. As the I/E weight ratio increases, 90

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3/11 the distributions of avalanche duration and size deviate from power-law distributions, 91 with large avalanches becoming less prominent. We use a previously developed measure, 92 called κ , to quantify how much a distribution deviates from a power-law distribution 93 with exponent [5,11,14]. If the distribution is close to the power-law distribution, then 94 κ is close to 1, which occurs for both avalanche duration and size distributions when 95 the I/E weight ratio is low. Any deviation in κ from 1 means a deviation from the  Considering the avalanche distributions and branching functions, we can confidently 120 conclude that our model operates near criticality for low I/E. Next, we sought 121 additional evidence to support the apparent possibility that high I/E corresponds with 122 the chaotic balanced state. Perhaps the most essential property of the chaotic balanced 123 state is that neurons should fire asynchronously; neurons should be weakly correlated. 124 Thus, we next examined the population-averaged input cross-correlogram (CCG) for 125 different I/E weight ratios (Fig 5a). When the I/E weight ratio is low, CCGs of 126 excitatory, inhibitory and total inputs are all high. As the I/E weight ratio increases,

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CCGs decrease. This decrease is most prominent at non-zero delays and for the total 128 input CCG. When the I/E weight ratio is high, although the excitatory and inhibitory 129 input CCGs remain relatively high at zero delay, the total input CCG is weak due to 130 tight balance between excitatory and inhibitory inputs. We quantify asynchrony based 131 on decreases in temporal and cross-neuron correlations. For this, we define η to be 132 inversely proportional to the area under CCG for total input (normalized as defined the 133 Methods). As shown in Fig 5b, asynchrony η sharply increases when the I/E weight 134 ratio goes beyond 2, and reaches a high value when the I/E weight ratios is near 4.

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Therefore, strong inhibition makes excitatory and inhibitory inputs balanced and leads 136 to an asynchronous activity. This is in accordance with the result in previous work that 137 the population-averaged firing correlation is weak when inhibition is strong and fast [17]. 138  21,32]. We next examined if this scaling rule emerges as we increased I/E 142 from low to high (see Methods). We found that for different I/E weight ratios, the 143 synaptic strength J as a function of K are power-law distributions with different scaling 144 exponents (Fig 6a). By curve fitting, we found a continuum of scaling exponents from 145 low to high I/E weight ratios (Fig 6b). As shown in Fig 6c, when the I/E weight ratio is 146 low, the exponent is near −1. When the I/E weight ratio is high, the exponent increases 147 to around −0.5. Thus, we conclude that at high I/E, our model confirms the 148 J ∼ 1/ √ K scaling expected for chaotic balanced networks. Considering together the 149 tight balance of E and I inputs (Fig 2), the asynchronous firing (Fig 5), and the 150 synaptic scaling (Fig 6), we conclude that high I/E in our model is consistent with the 151 chaotic balanced regime.

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We have shown that the population activity of neural networks can vary dramatically 154 depending how excitation and inhibition are balanced. If weak excitation is balanced by 155 weak inhibition, we found that the dynamics exhibit large fluctuations and rather 156 coordinated activity. If stronger inhibition balances stronger excitation in a higher 157 "tension" balance, we found that the dynamics are asynchronous and steady, consistent 158 with the chaotic balanced state. Our work establishes a simple bridge between two 159 previously discrepant views of cortical population dynamics. Our findings suggest that 160 the same network could be tuned from criticality to the chaotic balanced, by simply 161 strengthening inhibition and excitation. 162 One interesting hypothesis that emerges from our work concerns metabolic efficiency. 163 First, we note that maintaining a "strong" synapse depends on metabolically expensive 164 biophysical mechanisms -greater presynaptic vesicle pool, greater density of 165 postsynaptic receptors, etc. Since the high I/E regime and the low I/E regime have 166 similar firing rates, it stands to reason that the strong synapses of the high I/E scenario 167 would consume more metabolic resources than the lower I/E scenario. Moreover, the 168 critical dynamics we observed at low I/E are associated with a number of functional 169 benefits [10]. On the other hand, the lower fluctuations found in the high I/E regime 170 may be beneficial for functions that require lower "noise" [21,23]. When low noise is not 171 required, perhaps the brain could tune itself to the low I/E regime where energy 172 consumption is less. This is consistent with the observation that resting, awake animals 173 tend to exhibit greater fluctuations in population activity compared to alert, active We apply probabilistic integrate-and-fire dynamics on the recurrent network. The state 201 of each neuron is binary, either 1 or 0 corresponding to active or quiescent, respectively. 202 At each time step t, the probability of neuron i being active depends on two 203 independent factors: a probability p i due to synaptic inputs from other neurons within 204 the network and a probability p ext due to external inputs or spontaneous firing.
where s j (t − 1) is the state of neuron j at time step t − 1, and I i (t) represents the total 206 synaptic input to neuron i from other neurons at time step t. p ext is set as 0.005/N , where · indicates time average. Then, the E/I tension T of the recurrent network is 216 the average of T i over all neurons.

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The threshold for avalanches is determined at the level S * when the branching function 219 Λ(S * ) = 1.05, which is used in previous works [30]. We use κ to measure how much the 220 avalanche duration and size distributions deviate from power-law distributions [14]. κ 221 is defined as where F NA is the cumulative distribution function (CDF) of the reference power-law 223 distribution with exponent , and F is the CDF of the avalanche duration or size. From 224 the definition, κ is close to 1 if the measured distribution well matches the reference 225 power-law distribution, while either greater or smaller value of κ means deviation from 226 the reference power-law distribution. We use = 1.7 for avalanche duration and = 1.5 227 for avalanche size, which are the best-fitted exponents when there is no inhibitory 228 neurons. We take β i as a representative sample of 10 logarithmically spaced points 229 along the measured distribution.
where S(t) is the number of active neurons in the time step t. Since not all values of S 234 occur naturally in the models, we obtain a complete Λ(S) numerically, by simulating  Input cross-correlogram (CCG) 241 We plot the input CCGs by calculating the population-averaged cross correlations of 242 synaptic inputs with time lags from −20 to 20. To measure the desynchronization of the 243 total synaptic input from the synchronized excitatory and inhibitory synaptic inputs, we 244 define η as where A Total is the area under total synaptic input CCG, and A EE and A II are the area 246 under excitatory and inhibitory synaptic input CCGs, respectively.