Neuromodulatory control over nonlinear spiking of layer V pyramidal mediates adaptive slightly subcritical network dynamics Authors: Brandon R. Munn1,2,*, Eli J. Müller1,2, and James M. Shine1.

Affiliations

1 Brain and Mind Centre, The University of Sydney, Sydney, New South Wales, Australia

2 Complex Systems, School of Physics, The University of Sydney, Sydney, New South Wales, Australia.

* Correspondence to: [email protected].

Abstract To remain adaptable to a dynamic environment, coordinated neural activity in the brain must be simultaneously flexible and reliable. There is evidence that the brain facilitates this adaptive response using highly-conserved metabotropic neuromodulatory neurotransmitters, such as noradrenaline and acetylcholine. While we understand how these neuromodulators alter individual neuronal dynamics, precisely how operates in networks of neurons to give rise to observed large-scale dynamics remains unknown. By viewing neural dynamics as a critical phase transition, neuromodulators can be framed as control parameters that modulate the order inherent within a network of linear neurons – analogous to magnetic domains in the Ising model. While powerful, this view fails to account for two significant biological aspects: distinct arms of the neuromodulatory system possess differential anatomical projection patterns; and, neurons are not simple linear point sources. Neurons can spike nonlinearly, whereby the same input can elicit different responses. Here, we investigate this disparity by demonstrating correspondence between adaptive information processing modes – calculated on in vivo electrophysiological recordings of bursting layer V neurons in awake mice – and fluctuations in neuromodulatory tone – assessed by dynamic changes in diameter. We theoretically validate these results by creating a novel, biologically plausible dual-compartment model of nonlinear layer V pyramidal neurons – capable of both regular spike and bursting modes – that reproduce our main empirical findings. We then probe our model at a resolution impossible in vivo to demonstrate that the adrenergic and cholinergic neuromodulatory systems shift the brain into a neuroprotective (i.e., slightly subcritical) regime – assessed by the branching parameter – while facilitating flexible and reliable dynamics, respectively. This unexpected result demonstrates that the brain has circumvented the necessity to shift the system closer to criticality for variability by differentially augmenting an intrinsic neuronal nonlinearity. Our analyses establish that these distinct arms of the ascending arousal system modulate ensembles of layer V pyramidal neurons to augment critical dynamics and facilitate distinct adaptive information processing modes within the brain.

1 Introduction has been thoroughly investigated [7–9]. Briefly, this How to successfully overcome an uncertain, hypothesis posits that the brain is poised near the unconstrained, and dynamic situation – be that critical point [7], as these regimes are endowed with through continuing with the current strategy numerous beneficial properties, such as the ability to (reliability), switching strategies (flexibility), or strike an optimal balance between flexibility and assessing alternative strategies (sensitivity) – is a reliability, as well as the capacity to operate across fundamental problem faced by all living organisms. broad spatiotemporal scales [9–11] and optimize How the brain supports these different information information transfer between itself and the complex processing modes while retaining the ability to natural environment [9,12–14]. Using this approach respond with a high degree of specificity to distinct the collective dynamics of a system can be effectively ethological contexts remains an important open described by order parameters that are altered by question in neuroscience. By altering the excitability control parameters – for instance, in the well-known and receptivity of targeted neurons distributed Ising model, magnetism is an order parameter that is through the brain [1], the ascending altered by a control parameter of temperature [15]. neuromodulatory system is well-placed to solve this To the extent that the brain exhibits features of dilemma. The neuromodulatory system allows the criticality, as it appears that it does [9,16,17], we can brain to separate the control of the information then ask what features of the brain may act as control processing modes of the brain from the processing of parameters. specific bits of information. In this way, the brain can flexibly and adaptively transition between distinct In previous work, we have argued that the neural states that arise from dynamic endogenously neuromodulatory arousal system is well-placed to and exogenously driven demands in a manner that act as a control parameter in the brain [1,2,18,19]. The does not infringe upon the ongoing information arousal system is comprised of a set of neurons in the processing strategies. brainstem and that send a diverse set of axonal projections to the rest of the central nervous While there are many unique arms of the ascending system in unique ways that suggest unique arousal system [1], the concentration (Fig. 1A) of two functional consequences [2]. For instance, due to highly-conserved metabotropic neurotransmitters – diffuse projections of the locus coeruleus (Fig.1A noradrenaline (NAd) and acetylcholine (ACh) – has red), NAd is hypothesized to act as a global control been argued to transition the brain into distinct states parameter – like temperature in the Ising model. In with unique computational benefits [2]. For instance, contrast, the brain can simultaneously different fluctuating levels of NAd in the cerebral cortex have control parameters beyond this one-dimensional been associated with striking an optimal balance case, as cortical projections from the basal nucleus of between exploratory and exploitative behaviour [3], Meynert project in a much more targeted fashion whereas ACh levels are typically associated with (Fig.1A blue), suggesting that ACh acts as a spatially states of focussed attention [4]. Recent studies have localized control parameter preferentially altering demonstrated how these neuromodulators alter local coordination. individual neuronal activity [5,6]; nonetheless, how neuromodulation operates in networks of neurons to The site of action of neuromodulation is also of give rise to the observed emergent nonlinear crucial importance. Previous theories of neural dynamics remains poorly understood. criticality have treated individual neurons as linear point processes [12,16,20]. However, many neurons Concepts from physics are well suited for in the cerebral cortex are nonlinear – the same input understanding the collective phenomena of the can elicit different responses – suggesting that the brain [1]. In particular, the critical brain hypothesis neuromodulatory system could, in principle, confer

2 even more powerful computational benefits on the Neuromodulatory Mediated Signatures of brain than have been previously appreciated. The Criticality paradigmatic example of a nonlinear is the As a first step, we sought first to confirm evidence of thick-tufted Layer V pyramidal neurons (LVPN), the criticality in LVPN. To this end, we re-analyzed in vivo principal output neuron of the cerebral cortex and spontaneous electrophysiological neural activity capable of dual spiking modes [21–23]. LVPN possess recorded from three awake mice (Mus musculus) [28] two dendritic compartments – somatic and apical – collected from eight invasive silicon neuropixels that are electrotonically separated at rest by probes (Fig. 2A) in combination with pupil hyperpolarisation-activated cyclic nucleotide-gated recordings (Fig. 2B) [29]. These recordings provide (HCN) channels located along the dendritic apical high-resolution access to multiple cortical neurons in trunk (Fig. 1B). This separation ensures that inputs to awake animals. To isolate bursting neurons in Layer the apical dendrites do not typically affect spiking V, we applied two different criteria: first, we dynamics in the soma, which are instead driven selectively analyzed neuronal units that were solely by inputs close to the soma [24,25]. When the identified on channels referenced to Layer V of the dendritic compartments are coupled – e.g., via Allen Mouse Common Coordinate Framework [30]; closure of HCN channels or non-specific thalamic and second, we also identified units that drive – calcium spikes in the apical layer can demonstrated a bimodal ISI distribution, defined as propagate down the apical trunk allowing high- satisfying 푏퐼푆퐼/푛퐼푆퐼 > 10%, where 푏퐼푆퐼 is the number of frequency burst firing (Fig. 1C; [23]), during which ISI < 10ms and 푛퐼푆퐼 is the total number of ISI. This somatic sodium spikes ride atop the slower calcium approach ensured that we identified cells within spike [6]. In this way, LVPN spiking dynamics are Layer V that demonstrated both bursting (i.e., short nonlinearly dependent upon the state of the cell and ISI) and ‘regular’ spiking (i.e., long ISI) modes; (Fig. the broader network [26]. 2C). This procedure identified 148 Layer V bursting units across eight visual and motor sensory regions. Recent studies have confirmed that integral functions To support this claim, we further confirmed that the 푏퐼푆퐼 of the brain – such as conscious awareness and mean burst ratio of the isolated units (훽푚표푢푠푒 = = 푛퐼푆퐼 binding feedforward and feedback signals [6,22] – 0.26 ± 0.13 푠. 푑) was consistent with observed are coordinated through LVPN bursting. Significantly, bursting ratios of LVPN [23]. this bursting is modulated by adrenergic and cholinergic neuromodulation [5,6,27], suggesting To test our hypothesis, we also required a continuous that neuromodulatory control over LVPN may be measure of neuromodulatory tone, which is responsible for many computational benefits that this challenging to measure directly in the brain in vivo nonlinear mechanism might confer to the brain. without techniques like microdialysis (which was not Based on this work [5,6,27] and our previous available in this dataset) [31]. Despite this limitation, theoretical predictions [2], we hypothesize that ACh we were able to utilize fluctuations in pupil diameter and NAd act as orthogonal control parameters for as an indirect measure of neuromodulatory tone: the systems-level neuronal dynamics through the pupil dilates with increases in adrenergic tone (Fig. modulation of voltage-gated ion channels [1] on the 2B; left, red) and constricts with increases in apical dendrites of LVPN [27]. In the remainder of this cholinergic tone (Fig. 2B; left, blue) [31,32]. Based on manuscript, we provide both empirical and consternation in the literature, we re-analyzed the computational evidence that confirms these relationship between cortical NAd and ACh tone and predictions. pupil diameter in mice (see Supp.S1 for further details) [31]. Figure. 2B demonstrates that rapid Results: fluctuations in pupil diameter, 푝푑, (i.e., 푑푝푑/푑푡; Fig. Electrophysiological Evidence for

3 criticality in the empirical recordings, particularly via the increase in variability of the order parameter at the critical point - due to the divergence of the order parameters derivative [15]. To measure these properties in spiking data, we first calculated two distinct and commonly utilized neural order parameters: the population spike count,

1 𝜌 = ∑ 훿(푡 − 푡 ) , (1) 푡 푁 푖 푖

which represents the number of spikes at time 푡푖 Figure 1 – Adrenergic and Cholinergic across 푁 neurons [34] and the mean temporal neuromodulation target nonlinear layer V pyramidal coherence between neuronal spiking activity, neurons. A) The Locus Coeruleus (LC; red) and Basal Nucleus of Meynert (BNM; blue), which are 푁 predominantly responsible for cortical noradrenergic 1 휃 = ∑ | 푒푖휙푡푗 |, (2) (NAd) and cholinergic (ACh) metabotropic 푡 푁 neuromodulatory projections, contact layer V 푗=1 pyramidal neurons (LVPN) in a diffuse and targeted manner, respectively. B) Layer V pyramidal neurons where 휙푡 is the phase of the spike count obtained via (LVPN) in the cerebral cortex span all cortical layers and the Hilbert transform [20], and then we calculated the consist of two dendritic compartments (apical and somatic dendrites) that are electrotonically separated susceptibility as, by hyperpolarisation-activated cyclic nucleotide-gated (HCN) channels located along the apical trunk. C/D) 2 2 휒 = 〈𝜌푡 〉 − 〈𝜌푡〉 , (3) LVPN are capable of nonlinear spiking dynamics,

depending upon the stimulation location along the neuron’s dendrites. The electrotonic separation is such where 〈… 〉 represents the mean, quantifying the that the neuron typically undergoes regular spiking variability of the population spiking dynamics (Fig. (green; C) when driven by somatic drive; however, 2D blue) [9,35]. We also calculated the transient when the HCN-mediated electrotonic separation is exceeded within a short temporal window, the coherence as, simultaneous apical and somatic drive can switch the 2 2 neuron into a burst spiking mode (short inter-spike Ψ = 〈휃푡 〉 − 〈휃푡〉 , (4) interval [ISI]; yellow; D).

quantifying the variability in the coordinated spiking 2B right) are strongly related to the balance between phase (Fig. 2D yellow) [10,11]. Both measures track systemic adrenergic and cholinergic tone in the the spatiotemporal variance of order parameters of cerebral cortex, and hence can be used as an indirect the system – in this way, the diversity of readout of neuromodulatory tone in the brain. spatiotemporal spiking coordination can be

summarized to indicate the available information With these estimates of LVPN burst-firing and bandwidth of the system. For instance, large 휒 and Ψ neuromodulatory tone in place, we next required a suggests an extensive repertoire of utilized neuronal means for evaluating evidence of criticality. assemblies, which indicates a responsive and flexible Typically, we would quantify this as the proximity to information processing mode [20]. In contrast, small the critical point; however, due to the non- 휒 and Ψ indicate that variability is quenched into a stationarity of neuromodulatory tone, we cannot precise bandwidth, indicating a reliable information infer this directly from the dynamics [33]. processing mode [11,36]. Nevertheless, we can calculate signatures of

4 Using this approach, we set out to test the hypothesis nonlinear dynamic of the LVPN dual compartments – that NAd and ACh differentially affect systems-level or a combination of both. Nonetheless, the dynamic information processing dynamics [31]. We predicted that fluctuations in pupil diameter, 푑푝푑/푑푡, should differentiate distinct signatures of systems-level neural dynamics: phasic increases in NAd (i.e., pupil dilation; increased 푑푝푑/푑푡) should precede heightened 휒 and Ψ, whereas phasic increases in

ACh (i.e., pupil constriction; decreased 푑푝푑/푑푡) should precede decreases in both 휒 and Ψ. To assess these dynamic changes in both 휒 and Ψ with fluctuations in pupil diameter, we utilized time- varying estimates:

푑𝜌 휒̂ = | |, (5) 푑푡 푑휃 Ψ̂ = | | , (6) 푑푡 Figure 2 – Changes to neuronal dynamics in parallel

neuropixels electrophysiological recordings of LVPN in respectively. Our analysis confirmed our hypothesis: vivo under neuromodulation, consistent with specifically, we observed robust positive (negative) adrenergic and cholinergic neuromodulation relationships between pupillary dilation increasing and quenching variability, respectively. A & B) Under spontaneous conditions, spiking activity (A) 휒̂ < 7 × 10−6; (constriction; Fig. 2B) and both (p Fig. and time-varying pupillometry measures (B-left) and −5 2E) and Ψ̂ (p < 4 × 10 ; Fig. 2F). were recorded from 3 mice each across eight neuropixels distributed across the mice cortex. B-right) Adrenergic and cholinergic neuromodulatory tone were inferred Theories of neuromodulation hold that adrenergic via changes in pupil diameter, this approach is justified and cholinergic can shift the neuronal system as they are strongly correlated data obtained from [26] between flexible dynamics, in which neuronal (red line 95% CI that 푑푝푑/푑푡 and NAd-ACh are spiking is highly variable in both space and time, and uncorrelated, grey lines indicate zeroth). Note that the positive correlation suggests a positive change in pupil reliable dynamics, in which neuronal spiking diameter (dilation) corresponds to an increase in NAd patterns are more spatiotemporally constrained [31]. or decrease in ACh (adrenergic tone) and vice versa for Our results confirm that an increase in NAd (ACh) a pupil constriction (ACh >NAd). C) We isolated 148 tone leads to the enhancing (quenching) of the units recorded off channels located within Layer V that also possessed a bimodal ISI distribution suggestive of spiking variability and coherence of LVPN in both spiking and bursting dynamics (top-right). D) We recordings from an awake, freely behaving mouse in calculated the firing rate, 𝜌, of each Layer V unit and vivo. In this way, our results demonstrate a direct quantified the dynamics of the system by the spatial variability of coherence between individual neuronal relationship between adrenergic neuromodulation firing (Ψ yellow) and the temporal variability of the and flexible dynamics and cholinergic pooled population firing activity (휒 green). E & F) To neuromodulation and reliable dynamics at the calculate these measures at the rate of pupil fluctuations, network scale. we calculated the instantaneous approximation of the temporal variability of population spiking (휒̂; E) and spatial variability of coherence between neurons (Ψ̂; F). We found that adrenergic and cholinergic mediate Spatiotemporal variability increased during adrenergic signatures of criticality; however, we wondered neuromodulation (푑푝푑/푑푡 > 0; i.e., pupil dilating) and was quenched during cholinergic neuromodulation whether this was due to either changing the system’s (푑푝푑/푑푡 < 0; i.e., pupil constricting). proximity to the critical point or emerging from a

5 nature of neuromodulation during in vivo recordings coupled to one another via biophysical difference-of- and the difficulty from recording from both apical Gaussian (i.e., ‘Mexican-hat’) synaptic coupling and somatic compartments. Thus, to answer this weight 푤푖푗, question, we required a biologically plausible model of LVPN – with apical and somatic compartments – 0 if 푑푖푗 > 푑푚푎푥 or 푖 = 푗 2 2 푤 = { 푑푖푗 푑푖푗 } , able to reproduce the dual spiking dynamics. 푖푗 − − 퐶 푒 푑퐸 + 퐶 푒 푑퐼 if 0 < 푑 < 푑 퐸 퐼 푖푗 푚푎푥 (10)

A Biologically Plausible Network Model of where 푑푖푗 is the Euclidean distance between neuron 푖 Nonlinear Layer V Pyramidal Neurons and 푗, 퐶퐸 and 퐶퐼 are the excitatory and inhibitory Recapitulates Adaptive Dynamics Under coupling constants, and 푑퐸 and 푑퐼 are the excitatory Neuromodulation and inhibitory coupling ranges. This coupling has To discern the mechanistic origin of this critical been demonstrated to capture both local excitatory phenomenon, we constructed a biologically and inhibitory effects of LVPN [40]. Results presented plausible, spatially embedded neuronal network are for 푁2 = 4,900 neurons (a 70 × 70 grid) (Fig. 3A), in which model neurons consisted of two simulated for 푇 = 20s at a temporal resolution of compartments designed to mimic the dual 푑푡 = 0.5ms, and results were consistent for grid sizes compartment coupling of LVPN [24]. The somatic 푁 = 50 to 200 and 푇 = 5 to 50s (see compartment was modelled as an Izhikevich Supplementary for a detailed description). neuron [37] given by two coupled ODEs

푑푣 Typical neuronal modelling focuses on either the = 0.04푣2 + 5푣 − 푢 + 퐼, & (7) 푑푡 single neuron – at a multi-compartment resolution – 푑푢 or a network of neurons – each at a point-neuron = 푎(푏(푣 − 푣푟) − 푢), (8) 푑푡 resolution. By utilizing our novel dual-compartment

Izhikevich network model, we were able to create where 푣 represents the membrane potential (mV), 퐼 biologically plausible spike profiles while retaining represents all current into the neuron, 푎 = 0.02 ms−1 computational efficiencies (i.e., avoiding multiple represents the time constant of the spike adaptation channel kinetics across multiple compartments) [23]. current, 푏 = 0.2 nS describes the sensitivity of the In this way, we were able to examine the network- adaptation current to subthreshold fluctuations of level interactions of ensembles of thousands of LVPN the membrane potential, and 푣 represents the reset 푟 under a precise combination of neuromodulatory voltage, where the spike reset is control that would be impossible in vivo. 푣 ← 푐(푡) if 푣 ≥ 30 , then { }. (9) 푢 ← 푢 + 푑(푡) To systematically examine the effect of The somatic compartment is coupled to an apical neuromodulation on LVPN dynamics, we simplified compartment that acted as a temporal integrator our model parameter space to investigate two switch. The presence of coincident apical drive that independent parameters influenced by exceeded the apical-somatic electrotonic separation neuromodulation. Specifically, we investigated the within a 30ms window [38] caused the apical probability that apical drive exceeds the apical- compartment to switch the somatic spiking somatic HCN electrotonic separation leading to bursting, 훽 (Fig. 3B). This parameter captures any properties from empirically observed LVPN regular spiking (푐(푡) = −65, 푑(푡) = 8) to a burst firing effects that alter either HCN channel properties or the behaviour (푐(푡) = −55, 푑(푡) = 4; see Supp.S2 for full- excitability of apical dendrites. 훽 ranges from 훽 = details) [39]. The somatic compartments were 0 where LVPN cells cannot burst to 훽 = 1 where LVPN

6

Figure 3 - Biophysically plausible model of nonlinear Layer V pyramidal neurons under neuromodulation recapitulates slightly subcritical dynamics. A) We simulated a neuronal network of LVPN with apical and somatic compartments and isolated the effects of neuromodulation of the cells inherent nonlinearity by changing two parameters: the LVPN burst probability (훽) and the apical input spatial correlation (𝜎). B) Spiking data were simulated across a [훽, 𝜎] model parameter space, within which we could track the hypothetical trajectories of increasing neuromodulatory tone: NAd and ACh both increases bursting probability; however, NAd increases the apical input spatial correlation due to the LC diffuse projections, whereas cholinergic projections are known to be more segregated (Fig. 1A). The neuromodulatory trajectories are constrained within the empirically observed range of mouse Layer V bursting probability (purple bar mean ± s.d). C) Temporal spiking variability as quantified by the susceptibility (휒; thick lines = mean and thin lines = 95th confidence interval) and D) spatial variability as quantified by transient coherence (Ψ; error-bars are 95th confidence interval) peak following the adrenergic trajectory in parameter space (B) and v.v. for cholinergic tone (Measures follow the trajectories outlined in (B)). E) 푚 represents a measure of the proximity of a system to the critical point of a branching process – the panel depicts three typical regimes of a critical branching process: in the subcritical regime (i.e., 푚 < 1; left), activity is driven to quiescence; at the critical point (i.e., 푚 = 1; middle), activity is self-sustaining; and in the supercritical regime (i.e., 푚 > 1; right), activity increases over time. F) Under low neuromodulation, LVPN are near critical, and neuromodulation shifts the system further sub-critical away from the critical point. cells only burst. Second, we investigated the spatial influence of correlated apical drive, 𝜎 (smoothed by Based on their differential neuroanatomy, NAd and a 2-dimensional gaussian with s.d. = 𝜎; Fig. 3B). This ACh were hypothesized to have divergent effects on parameter captures differential profiles of apical 훽 and 𝜎. Both systems increase 훽, albeit through drive to the system, ranging from bursting LVPN cells different molecular pathways: NAd promotes LVPN that are spatially uncorrelated (i.e., cells are unlikely bursting via the alpha-2A receptor-mediated closure to be adjacent; 𝜎 = 1) to LVPN cells that are strongly of HCN-gated Ih channels [27]; and ACh by spatially correlated (i.e., cells burst adjacently; 𝜎 = depolarising M1 receptors [5]. Despite this similarity, 푁). We stimulated each parameter combination with the two systems have divergent effects on 𝜎 [2]: identical white noise drive to somatic and apical adrenergic projections are diffuse and cross-regional compartments – before spatial smoothing by 𝜎 – and boundaries in the cerebral cortex [42], whereas analyzed the emergent spiking dynamics. The cholinergic projections are typically more combination of nonlinear neurons and diffuse segregated [43] (Fig. 1A). For these reasons, in the coupling was sufficient to create substantial [훽, 𝜎] model parameter space, the effect of NAd is heterogeneity in the model's emergent spiking conceptualized as a right-upward trajectory (red dynamics. We also confirmed that an even mixture of arrows; Fig. 3B), whereas ACh is conceptualized as a spatially correlated bursting and regular spiking was left-upward trajectory (blue arrows; Fig. 3B). Note associated with an elevated, albeit low mean that other mechanisms (not considered here) may pairwise spike-count correlation (|푟푆퐶| < 0.07), which move the brain along distinct state-space trajectories, is consistent with experimental predictions [41]. including NMDA receptor engagement, which

7 increases apical excitability [44] (i.e., an upward lines corresponds to 95% confidence intervals across trajectory), or non-specific thalamic activity [45], 100 simulations). These modelling results provide which increases diffuse apical input and excitability robust confirmation of the empirical signatures of the (i.e., a right-upward trajectory). systems-level control of the ascending arousal system on the primary output cell of the cerebral cortex. Neuromodulation Shapes Critical Dynamics in Nonlinear Layer V Pyramidal Neurons Now that we have demonstrated the validity of the To date, theoretical work into criticality within nonlinear neuronal model, we can prove the neuronal networks has hypothesized that critical hypothesis that neuromodulation controls critical signatures emerge from complex interactions dynamics in nonlinear LVPN. To do so, we calculated between coalitions of relatively simple (i.e., linear) the branching process control parameter, 푚 [9]. The ‘point’ neurons [7]. This viewpoint has not yet branching process control parameter quantifies one embraced the abundance of inherently nonlinear spike’s likelihood to beget a second spike. We neurons within the mammalian brain (e.g., calculated this control parameter as,

LVPN [23,46], whose simple interactions may play a crucial role in the emergence of complex, adaptive 〈𝜌푡+1|𝜌푡〉 = 푚〈𝜌푡〉 + ℎ, (11) dynamics. With our computational model, we can now set the neuromodulatory tone as a constant, where the population spiking activity in the next time enabling us to calculate the proximity to the critical step 𝜌푡+1 is determined by the control parameter, 푚, point of the branching process – the universality class an external input ℎ, and 〈. |. 〉 is the conditional of neuronal avalanches [7,33]. expectation. 푚 signifies the phase of the system, wherein the critical point (푚 = 1) separates run-away We first ensured the model could reproduce our (푚 > 1; supercritical) from quiescent (푚 <1; empirical findings (Fig. 2E & F). We found that our subcritical) activity (Fig. 2E). As demonstrated in Fig. model could recapitulate the shifts in spatiotemporal 2F, the system is poised close to criticality (푚 ~0.98), variability of LVPN under both NAd and ACh albeit in a slightly sub-critical, ‘reverberating’ neuromodulation. Importantly, unlike in vivo regime [16,33]. A system poised in the reverberating experiments – which are fundamentally dynamic – regime can benefit from being near criticality while we were able to set the model to a constant mitigating its adversary offsets, such as balancing neuromodulatory tone and hence, calculate both sensitivity vs specificity [47], stimulus detection vs 휒 and Ψ on the spiking outputs of our model (Fig. 2D) discrimination [48], and integration time, which without requiring the dynamic 휒̂ and Ψ̂ increases with proximity to the critical point [11]. approximation (Fig. 2E & F). Using this approach, we constrained our NAd and ACh trajectories within the Unexpectedly, the addition of both NAd and ACh in empirical bursting regime observed in awake mice our model shifts the system into an even more (Fig. 3B, purple error-bars). We found evidence to subcritical regime, this is despite both NAd and ACh support the hypothesis that NAd and ACh increasing the bursting likelihood. Interestingly, this differentially affect the information processing finding contradicts the expected result that NAd dynamics of the cerebral cortex. Specifically, we should shift the system closer to criticality – thereby demonstrated that NAd increases spatiotemporal increasing variability – and vice versa for ACh. Our variability (Fig. 3C & D, increasing red) by engaging results indicate that the opposing adaptive modes of spatially correlated ensembles of bursting LVPN. In NAd and ACh emerge from their respective contrast, ACh was found to quench spatiotemporal anatomical projections – the globally controlling variability (Fig. 3C & D increasing blue) by engaging diffuse adrenergic neuromodulation promotes spatially separated ensembles of bursting LVPN (thin variability and flexibility, whereas the locally

8 controlling targeted cholinergic neuromodulation the width of the flashlight, respectively. A robust supports selectivity and reliability. Thus, our finding method for probing a system's information emphasizes the importance of studying processing capacity is to investigate its transfer nonlinearities in critical systems, as this phenomenon function (or input-output/gain curve), which maps a would not be present in coalitions of linear point- precise input to a characteristic output. In neurons [7]. Our results suggest that the spiking psychophysics, a typical transfer function fit to dynamics of LVPN in awake mice are poised near experimental data is the power-law function criticality and that neuromodulation can flexibly act F(S)~Sα, known as Stevens law, where S is the as a control parameter shifting the system from this stimulus intensity, 훼 the scaling ‘Stevens’ exponent, point to utilize the beneficial signatures of criticality and F(S) is the neural response to the stimulus [50]. while remaining in a neuroprotective slightly An efficient transfer function possesses an 훼 < 1 as subcritical regime. this allows a given range of stimuli to be mapped onto a smaller output range. We calculated the transfer function as,

푇 1 F(S) = ∑ 𝜌 (S) , (12) 푇 푡 푡=1

which is the mean spike density response, F(S), to

afferent Poisson spikes, 𝜌 , with a mean-rate S Figure 4 – Functional benefits of critical 푡 neuromodulation for information processing. randomly distributed across the network. A useful A) We analysed the information processing of the metric that can be calculated from the transfer neural system in the baseline neuromodulatory function are the dynamic-range, regime from Fig. 3B (purple) and two regions

corresponding to either a cholinergic (blue) or S0.9 ΔS = 10 log10 ( ) , (13) adrenergic (red) phasic burst. The transfer 푆0.1 functions between input stimuli intensity S and mean output F across repeated trials, reveals which represents the range of discriminable similarities and differences between the three stimuli [12]. The range [푆0.1, S0.9] are inverted from regimes. The dynamic range, ΔS, and trial-to-trial the transfer function [F0.1, F0.9] with Fx = F0 + variability, ΔF were also calculated. B) the low x(F − F ) F F arousal (purple) had the highest ΔS (i.e., the ∞ 0 where ∞ and 0 represent the saturation highest sensitivity) and moderate ΔF, whereas and baseline response, respectively. Another useful ACh (blue) minimised ΔF and thus increases measure is the trial-to-trial variability of the transfer specificity and signal reliability, whereas NAd function, (red) maximisedΔ and hence promoted a more F flexible information processing mode. Δ = 〈 푉푎푟(10 log F(S)) 〉, (14) F 10 Critical Dynamics Mediate Functional Information Processing Modes representing the intrinsic reliability and variability in Based on these results, we further hypothesized that mapping a stimulus to output [47]. cholinergic and adrenergic neuromodulation should differentially augment the network’s receptivity to To test our hypothesis, we calculated the transfer incoming stimuli [2]: NAd should augment function in three regions of parameter space, which flexibility and variability (albeit nonlinearly) [3], were chosen to represent the awake state (i.e., the whereas ACh should enhance reliability and base of the neuromodulatory trajectories in Fig. 3B; selectivity [49], analogous to widening or focusing Fig. 4A purple; 𝜎 = 39, 훽 = 0.2), as well as under

9 either adrenergic (Fig. 3B tip red arrow; Fig. 4A red; modulation could adaptively facilitate flexible and 𝜎 = 52.3, 훽 = 0.44) or cholinergic (Fig. 3B tip blue reliable information processing modes and arrow; Fig. 4A blue; 𝜎 = 25.7훽 = 0.44) somewhat unexpectedly shifted the network into a neuromodulation. We found the three transfer neuro-protective slightly subcritical regime functions all followed a power-law between their demonstrated by a branching parameter less than baseline and saturation values with the same scaling unity. Nevertheless, despite this subcritical shift exponent 훼~0.8, suggesting they efficiently map a away from the critical point, the interaction between large stimuli range to a smaller output, and that the adrenergic and cholinergic anatomical projections psychophysical-law is invariant to arousal state (i.e., and LVPN apical dendrites led to opposing beneficial equivalent differences in stimulus lead to a adaptive dynamics – increasing and decreasing 휒 and proportional change in perceived magnitude across Ψ, respectively, which are typically prescribed to a arousal). (Fig. 4A). It should be emphasized that one-dimensional critical phase transition [21–24]. In these properties are emergent phenomena and have this way, our results thus provide a natural not been coded into the network. The low resolution to the fundamental problem attributed to neuromodulation regime (purple) possessed the the critical brain hypothesis: namely, that while largest dynamic range, 훥푆, (Fig. 4B), consistent with criticality is beneficial for information its proximity to the critical point (Fig. 3F). Increasing processing [9,13,14,16,51,52], being poised near

NAd led to the largest trial-to-trial variability, 훥퐹, criticality is biologically risky, as it can lead to run- (Fig. 4B; red), which is consistent with the theory that away activity (such as seizures) [10]. NAd facilitates flexible behaviour (i.e., a diffuse flashlight beam; 2). In contrast, increasing ACh led to Through our findings, we can re-interpret canonical a reduction in variability (Fig. 4B blue), results from experimental neuroscience. For instance, corresponding to an increase in stimuli specificity the association of ACh with a heightened focus [53] and reliability, consistent with the known may relate to its capacity to constrain the variability enhancement of stimulus detectability and focus of spatiotemporal neural dynamics. Similarly, the with the increased cholinergic tone, i.e., a focused relationship between NAd tone and both cognitive flashlight beam [49]. function [3] and adaptability [54] can be reframed as arising from the augmentation of inherent Our findings thus present an optimal solution to the nonlinearities within populations of pyramidal cells inefficiency of subcritical dynamics and pitfalls of within the cerebral cortex. Furthermore, this result near-critical dynamics [47]: the unaroused (low provides insight into the large variability in trial-to- neuromodulatory) brain is associated with optimal trial responses to identical stimuli observed in animal signal detection (sensitivity), and two highly- recordings [55]. The slow-drift in response conserved neuromodulatory axes either sharpen variability likely corresponds to fluctuations in specificity and reliability (ACh) or widen the neuromodulatory tone, changing the system's neural flexibility of the system (NAd). gain. In particular, experiments have demonstrated quenched variability at the onset of stimulus Discussion trials [56], which in our model corresponds to an Our results demonstrated a biological increase in cholinergic tone. We predict that other implementation for controlling different information arms of the ascending arousal system, such as the processing modes in the brain – namely, by using dopaminergic, serotonergic, and histaminergic distinct arms of the ascending arousal system as systems (to name a few), will play similar roles as control parameters to mediate population-level, control parameters, albeit constrained by the unique slightly subcritical dynamics in nonlinear cortical circuits that these systems innervate. neurons. We found adrenergic and cholinergic

10 In conclusion, the confusion surrounding how biological neuromodulation can mediate beneficial information processing dynamics without pushing the system too close to the supercritical phase in the awake, mammalian brain can be rationalized by incorporating their anatomical and physiological properties. Whereby, despite both adrenergic and cholinergic neuromodulation increasing the bursting likelihood and shifting the system into a more subcritical regime – due to their differential anatomical intersection with the apical dendrites of

LVPN – they can adaptively mediate distinct signatures of criticality. Specifically, diffuse adrenergic projections promoted spatiotemporal neuronal variability, whereas targeted cholinergic projections quenched spatiotemporal neuronal variability. We demonstrated that a dual – local vs global – control of dynamics is an effective technique for efficient neural operation emphasized as the brain processes many critical biological functions, such as conscious perception and feedforward-feedback integration, through neuromodulatory altered LVPN. Thus, understanding how neuromodulation alters

LVPN neuronal dynamics is essential for the effective treatment of pathologies in cognitive function caused by failures of the neuromodulatory system, such as dementia, and incorporating this dual-controllability into other dynamical learning systems, such as neural network organization, will likely be beneficial.

Acknowledgments We would like to thank Matthew Larkum, Michael Breakspear, Russell A. Poldrack, Cristopher Whyte, Rick Shine, and Paul Martin for their constructive notes on the manuscript. The Harris Lab and Janelia for their public repository of Eight-probe Neuropixels recordings in three mice. JMS was supported by an NHMRC Investigator grant (#1193857) and a University of Sydney Robinson Fellowship.

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