THE STATES OF STATISTICAL MECHANICS
Cortical networks are shaped by the combined action of excitatory and inhibitory interactions. Among other important functions, inhibition solves the problem of the all-or-none type of response that comes about in purely excitatory networks, allowing the network to operate in regimes of moderate or low activity, between quiescent and saturated regimes. Here, we elucidate a noise-induced effect that we call "Jensen's force" -stemming from the combined effect of excitation/inhibition balance and
... bition balance and network sparsity-which is responsible for generating a phase of self-sustained low activity in excitationinhibition networks. The uncovered phase reproduces the main empirically-observed features of cortical networks in the so-called asynchronous state, characterized by low, un-correlated and highly-irregular activity. The parsimonious model analyzed here allows us to resolve a number of long-standing issues, such as proving that activity can be self-sustained even in the complete absence of external stimuli or driving. The simplicity of our approach allows for a deep understanding of asynchronous states and of the phase transitions to other standard phases it exhibits, opening the door to reconcile, asynchronousstate and critical-state hypotheses, putting them within a unified framework. We argue that Jensen's forces are measurable experimentally and might be relevant in contexts beyond neuroscience. Networks of excitatory units -in which some form of "activity" propagates between connected nodes-are successfully used as abstract representations of propagation phenomena as varied as epidemics, computer viruses, and memes in social networks 1 . Some systems of outmost biological relevance cannot be, however, modeled simply as networks of excitatory units. Nodes that inhibit (or repress) further activations are essential components of neuronal circuits in the cortex 2 , as well as of gene-regulatory, signaling, and metabolic networks 3,4 . Actually, these are an essential feature of cortical networks as synaptic excitation occurs always in concomitance with synaptic inhibition. What is the function of such a co-occurrence? or, quoting a recent review article on the subject, "why should the cortex simultaneously push on the accelerator and on the brake?" 5 . Generally speaking, inhibition entails much richer sets of dynamical patterns including oscillations and other counterintuitive phenomena 6,7 . For example, in a nice and intriguing paper that triggered our curiosity, it was argued that inhibition induces "ceaseless" activity in excitatory/inhibitory (E/I) networks 8 . More in general, inhibition helps solving a fundamental problem in neuroscience, namely, that of the dynamic range, defined as follows. Each neuron in the cortex is connected to many others, but individual synapses are relatively weak, so that each single neuron needs to integrate inputs from many others to become active. This leads to the existence of two alternative phases, a completely quiescent and an active/saturated one (phases are characterized by the average value of the network activity which acts as a control parameter). In other words, increasing the synaptic coupling strength leads to an explosive, all-or-none type of recruitment in populations of purely excitatory neurons when a threshold value is crossed, i.e. to a discontinuous phase transition between a quiescent and an active phase 5 . Having only two possible phases (quiescent and active/saturated) would severely constrain the set of possible network states, hindering the network capacity to produce diverse responses to differing inputs. This picture changes dramatically in the cortex, where the presence of inhibition has been empirically observed to allow for much larger dynamic ranges owing to a progressive (smoother) recruitment of www.nature.com/scientificreports www.nature.com/scientificreports/ neuronal populations 9,10 . This is consistent with the well-known empirical fact that neurons in the cerebral cortex remain slightly active even in the absence of external stimuli 11-13 . In such a state of low self-sustained activity neurons fire in a steady but highly-irregular fashion at a very low rate and with little correlations among them. This is the so-called asynchronous state, which has been argued to play an essential role for diverse computational tasks     . It has become widely accepted that such an asynchronous state of low spontaneous activity emerges from the interplay between excitation and inhibition. Models of balanced E/I networks, in which excitatory and inhibitory inputs largely compensate each other, constitute -as it was first theoretically proposed 18-22 and then experimentally confirmed 23-27 -the basis to rationalize asynchronous states. Indeed, balanced E/I networks are nowadays considered as a sort of "standard model" of cortical dynamics 28 . In spite of solid theoretical and experimental advances, a full understanding of the phases of E/I networks remains elusive. For instance, it is still not clear if simple mathematical models can sustain highly-irregular low-activity phases even in the complete absence of external inputs from other brain regions. Indeed, many existing approaches to the asynchronous state assume that it requires of external inputs from other brain regions to be maintained 29 , while some others rely on endogenously firing neurons -i.e. firing even without inputs-for the same purpose (see e.g. 30 ). Furthermore, it is not clear from modelling approaches whether asynchronous states can have very low (rather than high or moderate) levels of activity 29,31,32 . All these problems can be summarized -from a broader Statistical Mechanics perspective-saying that it is not well-understood whether the asynchronous state constitutes an actual physical phase of self-sustained activity different from the standard quiescent and active ones. It is not clear either if novel non-standard types of phase transitions emerge at its boundaries. Such possible phase transitions might have important consequences for shedding light in to the so-called "criticality hypothesis". This states that the cortex might operate close to the edge of a phase transition to optimize its performance. To shed light onto such a conjecture it is essential to first understand what the possible phases and phase transitions of cortical networks are. Here, we analyze the simplest possible network model of excitatory and inhibitory nodes in an attempt to construct a parsimonious -understood as the simplest possible yet not-trivial-approach to of E/I networks 8 . We show, by employing a combination of theoretical and computational analyses, that the introduction of inhibitory interactions into purely excitatory networks leads to a self-sustained low-activity phase intermediate between conventional quiescent and active phases. Remarkably, the novel phase stems from a noise-induced mechanism that we call "Jensen's force" (or "Jensen's drift") -for its relationship with Jensen's inequality in probability theoryand that occurs owing to the combined effect of inhibition and network sparsity. The low-activity intermediate phase shares all its fundamental properties with asynchronous states and thus, as we argue, our model constitutes the simplest possible statistical-mechanics representation of asynchronous endogenous cortical activity. Moreover, continuous (critical) phase transions -separating the novel intermediate phase from the quiescent and active phases, respectively-are elucidated, with possible important consequences to shed light on the criticality hypothesis 33-35 , and to make an attempt to reconcile the asynchronous-state and criticality hypotheses, putting them together within a unified framework. Finally, we propose that the elucidated Jensen's force might be relevant in other contexts such as e.g. gene regulatory networks.