Abstract
Persistent activity is thought to mediate working memory. While such stimulus evoked persistence is well studied, mechanisms of internally generated or spontaneous persistence in vivo are unknown. Further, current theories based on attractor dynamics focus on elevated activity as a memory substrate, while little attention has focused on the role of inactivity attractors. Here, we present a mean field model of functional interaction between large cortical networks that predicts both spontaneous persistent activity (SPA) and inactivity (SPI); the latter has never been seen before in experiments or models. We confirm these predictions using simultaneously recorded neocortical local field potential (LFP) and the membrane potential (Vm) of identified excitatory neurons from several brain areas in vivo during slow oscillations, especially from layer 3 of the medial (MECIII) and lateral entorhinal cortex (LECIII), which show SPA and SPI. By matching model and experimental statistics, we predict the relative strength of internal and external excitation in the LECIII and MECIII networks. Our predictions match anatomical data. Further, the model predicts, and the experiments confirm, that SPA and SPI are quantized by cortical UDS and follow the statistics of a history dependent Bernoulli process. These convergent, theory-experiment results thus reveal the differential nature of cortico-entorhinal functional connectivity, resulting in a unique pattern of persistent activity and persistent inactivity, a novel and energetically efficient memory substrate.
Introduction
Cognition requires the interaction between several neural networks, each network containing millions of neurons, each neuron in turn characterized by many microscopic parameters. To study the complex emergent properties of systems with large degrees of freedom, the statistical physics approach is to develop a quantitative model, based on only the salient order parameters, and subsequently test its predictions in simplified experimental preparations that capture the essence. In line with this tradition, we develop an analytically tractable model of spontaneous activity in interacting neural networks, and quantitatively verify several predictions of the theory in vivo during default, internally generated activity in the absence of external sensory stimuli.
During quiescence, deep sleep, under anesthesia, and in vitro, local neural networks from many brain areas, including cortex, show synchronous, rhythmic activity termed delta oscillations, non-REM sleep oscillations, slow wave sleep (SWS) etc.1–4. The LFP shows rapid transitions between periods of elevated activity (the Up state) and silence (the Down state). The membrane potential (Vm) of individual neurons exhibits synchronous transitions between depolarized (Up) and hyperpolarized (Down) states. These Up-Down states (UDS) are ubiquitously found across species and experimental preparations, and are considered the default activity of many networks5–8. Several studies have suggested that the interactions between cortical regions during UDS are crucial for memory consolidation9–13. Impairment of UDS causes learning and memory deficits, while UDS enhancement leads to improvement14,15.
Although most cortical areas show synchronous UDS oscillations11, recent studies have shown that in vivo only MECIII, but not LECIII, pyramidal neurons show spontaneous persistent activity (SPA) during UDS: events where the neuron’s Vm persists in the depolarized Up state while the afferent neocortical areas transition to the Down state16. This definition notably differs from other studies that define singular Up states within an isolated network as themselves forms of persistent activity17. Instead, it is reminiscent of activity sustained after the extinguishing of a stimulus, hypothesized to form the neural representation for working memory during awake behavior. Existing network models show that such sustained activity during awake, working memory tasks can be generated through reverberant excitation and feedback inhibition, but it is unclear whether these models can explain spontaneously evoked persistent activity18,19. Depolarizing current injections do not elicit SPA within MECIII neurons during UDS, implicating network rather than intracellular mechanisms16. Existing network models of UDS employ an attractor framework with two fixed points, one active (the Up state) and one inactive (the Down state), with adaptation driving the oscillation20–24. Such models, however, have not been used to understand large interacting networks, and thus cannot account for major experimental findings, like the quantization of SPA during UDS25. Furthermore, existing theories focus exclusively on the active state, discarding the inactive state as simply a recovery phase for network adaptation. The energy function of discrete Hopfield networks26,27, however, is symmetric under activity inversion (+1 → −1), so the physics suggests these inactive states are themselves energy minima in the landscape and could thus also be utilized as a memory substrate.
We found that a simple, mean-field model involving two interacting networks of excitation-inhibition can capture the dynamics of SPA during UDS. Our theory also exploited the symmetric inactive attractor to predict a new phenomenon: spontaneous persistent inactivity (SPI). To test the model quantitatively, we used the in vivo cortico-entorhinal circuit as our model system. Anatomically, the neocortex serves as an afferent source of input to other cortical regions like the entorhinal cortex28,29. To measure neocortical ensemble activity during UDS, we recorded the extracellular LFP from the parietal cortex. As the parietal cortex receives strong inputs from neocortical regions30–33 and UDS is synchronous across all neocortical areas2,11,34,35 this LFP acted as the afferent reference for neocortical UDS. Simultaneously, we did wholecell Vm measurements from anatomically identified pyramidal neurons in various efferent areas, including parietal (PAR) and entorhinal cortices (EC); as the spontaneous activity of single neurons is tightly linked to the cortical networks in which they are embedded, this allowed us to probe the activity of localized networks within each target region36. Within the EC, the medial (MEC) and lateral (LEC) subdivisions are anatomically and functionally distinct: the MEC contains spatially selective “grid cells”37, while the LEC is thought to encode objects or experienced time38–43. We focused in particular on the EC layer 3 regions, since MECIII neurons are a major source of input to the hippocampus, show SPA in vivo, and are crucial in the generation and maintenance of UDS in the MEC16,44.
We detected in vivo SPA in MECIII and SPI in both MECIII and LECIII, but not in PAR. Further analysis of these events showed clear agreement with theoretical predictions, as both SPA and SPI were quantized by neocortical UDS and reflected the statistics of a history-dependent Bernoulli process. Our model attributed the differences in SPA and SPI across cells to differences in excitatory connectivity between the large neocortical network and the specific efferent subnetwork. The number of experimental observations explained by our model are greater than the number of parameters we varied, demonstrating its predictive power. To our knowledge, our study is the first to predict theoretically and detect experimentally the novel phenomena of persistent inactivity, and show that both SPA and SPI not only cooccur but are the result of common network interaction principles.
The mean field model of cortical interaction predicts both spontaneous persistent activity (SPA) and inactivity (SPI)
A minimal mean field network supporting UDS has three biologically well-established ingredients: excitation, inhibition, and the adaptation of excitation (but not inhibition)20–22,24. We constructed a mean field model of two cortical regions, each with its own recurrently connected inhibitory and excitatory populations45,46 (Fig 1). In isolation, each network exhibits transitions between Up and Down states (Sup. Fig 1-2) that are the stable fixed points of the dynamical system of equations, much like local minima in an energy landscape47,48. Their stability is inversely related to their distance from the separatrix, a line which defines the boundary between the basins of attraction. The slowly-varying, activity-dependent adaptation translates the excitatory nullcline, thus influencing the stability of each state. Growing adaptation governs the transition from the Up to Down state, while external drive and a falling adaptation governs the transition from Down to Up. Underlying gaussian noise gives the network a “temperature,” preventing it from stagnating in a particular state for arbitrarily long time periods. For simplicity, quantitative falsifiability, and based on available observations, we assumed that all internal parameters except the recurrent excitation strength WINT are identical across the afferent and efferent networks.
These two networks are connected unidirectionally, with the afferent network sending an excitatory projection WEXT to the excitatory population in the efferent network (Fig 1A, Sup. Fig 3). The UDS oscillation of the afferent network rhythmically destabilizes the endogenous UDS oscillation of the efferent network. While larger values of WEXT lead to phase locking (Fig 1C), smaller values give rise to transient desynchronizations between the two networks49. Under weak drive WEXT from the afferent network, an increase in the efferent recurrent excitatory connectivity (WINT) drives the efferent Up state fixed point away from the separatrix, increasing the Up state stability. Simulations demonstrate a novel, network mechanism to generate SPA: instances when the efferent network remains in the Up state, skipping one or more afferent Down states (Fig 1D). These could explain the experimentally reported SPA. Further, a decrease in the strength of WEXT decreases the destabilizing effect of the afferent transitions on the efferent network; the efferent then remains in a Down state, skipping one or more afferent Up states. We call this novel phenomena spontaneous persistent inactivity (SPI: Fig 1E). The model further predicts that SPA and SPI are relatively independent, as increasing WINT while simultaneously decreasing WEXT gives rise to coupled UDS sequences exhibiting both SPA and SPI (Fig 1F).
Detection of SPA and SPI in MECIII and LECIII
To monitor network interactions during spontaneous activity in vivo, mice were lightly anesthetized with urethane to induce robust and steady UDS that were synchronous across the entire neocortex. A hidden Markov model was used to classify the data into a binary UDS sequence50. Consistent with previous studies, the neocortical LFP and the Vm of neurons in PAR (N=24) and the efferent regions MECIII (N=50) and LECIII (N=16) showed clear bimodal UDS (Fig 2, Sup. Fig. 4). For subsequent analysis, the amount of SPA (SPI) was defined as the proportion of efferent Up (Down) states which outlasted an entire afferent Down (Up) state during an entire experiment. As a first test of the model, we computed the relationship between the neocortical LFP and the Vm from PAR pyramidal neurons, which were recorded close by (0.5 mm apart). Here, WEXT is large, and the model predicts complete phase locking (Fig 1C), with virtually nonexistent SPA and SPI. This was indeed the case (Fig 2B). Additional Vm measurements from neurons in frontal and prefrontal cortex also showed complete phase locking, consistent with WEXT from the neocortex being large (Sup. Fig. 4)11,30–35.
Consistent with previous studies, MECIII neurons showed clear instances of SPA (Fig 2C), while LECIII neurons did not16. In contrast, both LECIII and MECIII neurons showed clear instances of the newly predicted SPI (Fig 2D). Our model also predicted relative independence of SPA and SPI; consistently, some MECIII neurons showed both SPA and SPI, only a few seconds apart (Fig 2F), and levels of SPA and SPI within the population of LECIII and MECIII neurons were not significantly correlated (Sup. Fig. 5). Finally, SPA and SPI levels were not correlated with the duty cycle and the frequency of neocortical UDS, indicating that they were not artifacts of differences in brain states across experiments (Sup. Fig. 6).
Fitting experiment to model reveals differential connectivity within MECIII and LECIII
The properties of SPA and SPI not only varied across brain regions, but even between different neurons from the same region. We hypothesized that all of these differences could arise from just two network parameters: the strength of recurrent excitation in the efferent network (WINT) and the strength of external excitatory input to the efferent network (WEXT). To test this idea, we used a two-step approach. First, we simulated all possible networks in this 2D parameter space by varying only WEXT and WINT, while leaving all other variables unchanged. Modulating just two free parameters yielded networks with a wide range of both SPA and SPI. Thus, we could estimate the two crucial network variables, WINT and WEXT, by simply computing the amount of SPA and SPI observed experimentally (Fig 3A). Crucially, we did not match any other properties of SPA and SPI between the model and data. The robustness of this procedure was confirmed by using an alternate fitting procedure, which yielded very similar fits between the simulations and in vivo data for each neuron (Sup. Fig 7). Overall, a decrease in WEXT corresponded with higher levels of SPI, and an increase in WINT corresponded with higher SPA, as predicted by dynamical systems analysis (Fig 3A, B). While SPA and SPI prevalence across neurons was uncorrelated, the fitted values of WINT and WEXT were significantly negatively correlated, especially for LECIII, indicating differential properties of the networks (Sup. Fig 5).
The two parameter model, thus constrained by experiments, made major predictions about the nature of large-scale connections between and within these brain regions. Briefly, our model implies that neocortical input into MECIII is weaker than into LECIII, and still weaker than into other neocortical regions, like parietal, frontal, and prefrontal cortices. Further, it predicts that recurrent excitation within MECIII is stronger than within LECIII. These statements are corroborated by established experiments in vivo and in vitro (see Discussion). Additionally, several further predictions of the model could be tested using the match between experiment and simulation.
Inferred network connectivity predicts differential latency to UDS transitions in MECIII and LECIII
Since neurons behave like leaky capacitors, the strength of afferent excitatory input should be inversely correlated with the response latency of the efferent neurons51,52. Therefore, the model predicts that the neurons with larger values of estimated WEXT should respond sooner to neocortical Down-Up transitions, i.e. smaller latency between neocortical LFP and the neuron’s Vm (Fig 3C). Indeed, LECIII cells with greater predicted excitatory input WEXT showed significantly shorter Down-Up transition latency (Fig 3D-E). A similar result was found within the population of MECIII neurons. Further, consistent with model prediction that WEXT from the neocortex to LECIII is stronger than to MECIII, the population of LECIII neurons showed shorter Down-Up latency than the MECIII population (Fig 3D-E). While WEXT enhances the coupling between the two networks, larger values of WINT make the efferent network more independent of the input. The effect of these competing inputs is state dependent, differentially modulating the efferent Down-Up vs. Up-Down transitions. During an afferent Down-Up transition, the efferent network is in the Down state, where recurrent excitation WINT does not contribute. Thus, the latency of the efferent Down-Up transition should be relatively insensitive to WINT but depend strongly on WEXT. This was strongly supported across both LECIII and MECIII populations (Sup. Fig. 8).
The situation is reversed for the Up-Down transition, when the efferent network is in the Up state, where recurrent excitation WINT contributes strongly and helps sustain the Up state despite the loss of afferent input, which is in the Down state. Networks with higher WINT have more stable Up states, thereby increasing their “inertia.” Thus, the model predicts that ECIII neurons with greater predicted WINT should follow the neocortical Up-Down transitions with longer latency. This was confirmed for both MECIIII and LECIII (Fig 3F-G). In contrast to Down-Up transitions, the latency of the efferent Up-Down transition should be relatively insensitive to WEXT compared to WINT. This prediction too was supported across individual neurons within MECIII, within LECIII, and across the MECIII vs LECIII ensemble (Sup. Fig. 8).
These latencies were more correlated with the predicted WINT and WEXT values than with simply the levels of SPA or SPI (Sup. Fig. 8), further supporting the model. Additionally, neurons with greater net excitatory input (WEXT + WINT) should have higher firing rate; this was confirmed by experiments, showing greater mean firing rates for MECIII than LECIII, even at the level of individual cells (Sup. Fig. 9). Further, LECIII neurons’ Vm was significantly less depolarized than MECIII neurons (Sup. Fig. 9). The predicted model parameters WEXT and WINT were more strongly correlated with the UDS latencies than with the mean firing rates, further supporting the model and ruling out nonspecific effects.
SPA and SPI are quantized by neocortical UDS
The model predicts that SPA and SPI are all-or-none events that are initiated and terminated by state transitions in the afferent network. As a result, even though efferent Up(Down) state and SPA(SPI) durations form a continuous, unimodal distribution, these durations should be quantized in integral units of the afferent UDS cycles (Fig 4A, Sup. Fig. 10). To visualize this for SPA, segments of the simulated efferent activity were extracted around each efferent Down-Up transition, sorted according to the ensuing Up state duration, and assembled into a single matrix, with each row corresponding to a single efferent Down-Up transition (Fig 4B). The underlying afferent activity matrix for the same time points exhibited alternating bands of UDS, with integer multiples of afferent UDS fitting inside each efferent Up state (Fig 4C). The same visualization with in vivo data matched strikingly well with model predictions (Fig 4D). We repeated this for efferent Down states and SPI, yielding a similar quantitative match between the model and experiment (Fig 4E-F). When consolidating the rescaled state durations over all experiments and their matched simulations, the probability distributions for both were significantly multimodal, with peaks at half integers, indicating that ECIII state transitions were locked to the neocortical transitions, and that the ECIII skipped entire neocortical Up/Down states in integer quantities (Fig 4G, Sup. Fig. 11).
The multimodality of quantized durations was also observed for individual experiments and their corresponding simulations (Fig 5A, Sup. Fig. 11). We leveraged this distribution, unique to each cell, to investigate the precise history-dependence of SPA and SPI, further testing our model. One can imagine three scenarios. First, the SPA and SPI are entirely stochastic, in which case their probability distribution would follow a memoryless Bernoulli process, like a sequence of coin flips. Second, SPA and SPI arise due to some change in the overall state of the animal, such that all the SPA and SPI co-occur. However, our model predicts a third possibility: it should be rarer to have consecutive sets of SPA and SPI compared to singular events. This is because the probability of SPA and SPI is strongly history-dependent. If the network exhibits SPA at a given afferent Down state, the efferent network’s recurrent excitation WINT would be more adapted than usual, reducing the resources needed to sustain SPA in the next Down state, thus reducing the probability of consecutive SPAs. Similarly, the occurrence of SPI at a given afferent Up state would make the efferent network less adapted and hence reduce the probability of consecutive SPIs. To test this prediction, we used the first two modes of the quantized probability distribution (in Fig 5A) to calculate a1, the probability of a solitary SPA and SPI, and a2, the probability that another SPA and SPI occurred given a1 already happened. Here, a2=a1 for the first memoryless hypothesis, a2 > a1 for the second brain-state dependent hypothesis, and a1 > a2 for the third hypothesis, predicted by our model. The experiments strongly corroborated our predictions: the probability of SPA and SPI diminished after the first such event (Fig 5B). Thus the two network system has a “memory” of SPA and SPI due to the adaptation of the recurrent excitation WINT in the efferent network.
Discussion
Persistent activity has been hypothesized to mediate working memory via reverberating activity26,53, and has been studied extensively in vivo54–56, in vitro57–60, and in silica18,19. Its ubiquity and diversity in different cell types, brain regions, brain states, and behavior supports the hypothesis that a common mechanism could apply, and a low dimensional theory could be well suited to explain it. We developed a mean field model to explain the recent discovery of spontaneous persistent activity in MECIII during sleep16, as existing models focus only on stimulus evoked persistent activity during awake behavior. Using two networks of excitation-inhibition neurons and adapting excitation, with an afferent network providing excitatory input to an efferent one, our model reproduced phase locked Up-Down state (UDS) oscillations and the reported spontaneous persistent activity.
Further, the model exploited the symmetry of the discrete attractor landscape to make a surprising prediction, namely the existence of persistent inactivity. In contrast to persistent activity, which involves the efferent network sustaining activity while afferent inputs have shut off, persistent inactivity involves the efferent network sustaining inactivity while afferent input turns on. This has not been reported before in any experimental or theoretical studies, though there are hints61,62. Computational studies have found coexisting Up and Down states in different neurons within the same spiking network63,64, but these results are usually achieved when the network UDS is highly irregular and asynchronous. To test our model, we focused on the cortico-entorhinal interaction during UDS oscillations, using simultaneous LFP from the neocortex that served as a common afferent reference, along with the membrane potentials measured from anatomically identified neurons in the parietal, frontal, prefrontal, and entorhinal cortices.
The experiments confirmed the presence of both persistent activity and inactivity; we were then able to leverage these two observables to probe the underlying network architecture. Our framework models different brain regions by varying only two biologically relevant parameters: the strength of internal connections WINT within the efferent network and the strength of external input WEXT from the neocortex while leaving all the other parameters unchanged. Dynamical systems analysis47,48 showed that SPA increases with WINT, while SPI decreases with WEXT; thus, each cell, and the local network in which it is embedded36, could be mapped to the WINT-WEXT parameter space. Our results predicted that neocortical input onto the entorhinal region should be weaker than to other regions within the neocortex, like parietal, frontal, and prefrontal cortex. This is consistent with anatomical observations of strong intra-neocortical connections and weaker neocortical-entorhinal connections29,30,32,65,66. Within the entorhinal region, the model predicted that neocortical input into LECIII was significantly stronger than to MECIII. This is consistent with classic anatomical studies67,68, which show that a higher proportion of LEC afferents originate in cortical areas compared to MEC afferents, and more recent work40,69 showing stronger projections from the orbitofrontal cortex, part of the prefrontal cortex, to LEC compared with MEC.
Our analysis found greater amounts of persistent activity in MECIII than LECIII, and the model predicted that this is because the recurrent connections WINT should be larger within MECIII than within LECIII. This is indirectly supported by recent experiments showing greater recurrent connectivity between principal neurons within MECIII than within other MEC layers, and that MECIII network is crucial in the initiation and maintenance of the Up state during UDS in vitro in isolated EC slices44,70. Furthermore, excitatory cholinergic receptors are crucial for MECIII persistent activity71, and the application of acetylcholine to MEC slice preparations in vitro causes prolonged Up states in individual cells due to increased overall excitation and more frequent and rhythmic population-wide events, consistent with our hypothesis that persistent Up states are the result of networks having increased internal excitation WINT72.
The model with above network connectivity not only predicted the prevalence of SPA and SPI in the efferent neurons but also predicted their relative timing to afferent neocortical activity51, at both population-wide and single-cell resolution. Cells with higher predicted WEXT, and thus stronger coupling, exhibited significantly shorter state transition lags, while larger recurrent excitation WINT, and thus stronger “inertia,” had longer lags, as expected. The latency patterns were quite different for Down-Up vs. Up-Down transitions: the former was more dependent on WEXT, and the latter more on WINT. Our results thus support the hypothesis that the Up state is terminated by internal network mechanisms but is initiated by external input20,73. Taken together, these mechanisms resulted in systematic differences in the response latencies of MECIII and LECIII neurons during Up and Down states, which would influence the information processing in downstream hippocampal neurons11,16 and hence the memory consolidation process via spike timing-dependent plasticity mechanisms13.
As a direct consequence of the underlying physics of the model, we predicted that both SPA and SPI durations, while showing continuous, long-tailed distributions, should also show quantization in the units of afferent neocortical UDS cycles. This too was verified experimentally, with not just qualitative but a quantitative match between the model and experiment. Our model went further to predict that SPA and SPI were highly history-dependent, reducing the probability of consecutive SPA and SPI, and this too was confirmed in vivo. This long time-scale memory is an emergent property of the adaptation in the efferent EC network, which has been implicated in the formation and maintenance of periodic spatial firing of grid cells in MEC74.
While persistent activity has been studied extensively as the mechanism underlying working memory, it is far more energetically expensive than persistent inactivity. Furthermore, the models involving only persistent activity have a limited storage capacity, especially when dealing with memories that require overlapping representations75–77. Persistent inactivity introduces a new mechanism to overcome this difficulty. From an information theoretic perspective, a 0 is just as informative as a 1. Hence, a combination of persistent activity and inactivity would be an energy and information efficient scheme for storing overlapping memories by multiplexing the representation78,79. Related, our model predicted that the same neuron can show SPA and SPI, and this was experimentally confirmed. Recent theories investigated “persistent activity-silent” mechanisms for working memory and hypothesized that the information is stored in facilitated synapses80–82. One prediction is that non-specific inputs can reawaken the memory ensemble after the inactive period. Our model predicts, and experiments confirm, something similar: that the efferent network is more susceptible to inputs after SPI due to falling adaptation. These dynamics between adaptation and activity could drive the production of sequences of memories in neural networks with discrete83 and continuous phase spaces84.
The long duration of UDS under anesthesia allowed unequivocal detection of both SPA and SPI. But, since SPA and SPI remained unchanged across a range of anesthesia depths, and SPA has been shown in MECIII during drug-free sleep, these results should be broadly applicable16. On the other hand, a large number of biological factors that we did not consider could modulate our system wide findings. For example, in addition to the direct inputs from the parietal cortex to EC, there is substantial indirect input via the perirhinal and postrhinal cortices that we did not consider85. Recent studies show some cortical inhibitory neurons that remain active during the down state, which can alter the nature of cortical UDS86. Finally, hippocampus receives EC input and projects back to EC, and EC projects back to the frontal cortices; these connections were not included in our model, but could be studied in the future87. Despite this, the simple model was able to predict and match a large amount of experimental observations in a quantitative, cell-by-cell manner. Future studies can build on this approach to study SPA and SPI during drug-free sleep.
Given the direct and indirect pathways linking the entorhinal region to the hippocampus40,87, the decoupling of entorhinal activity from neocortical inputs during SPA and SPI could contribute to selective removal, strengthening, and weakening of memory traces from the hippocampus during slow-wave sleep, thus improving the signal to noise ratio in the space of memories, thereby improving experimentally observed task-related performance9. Our model is sufficiently general and could equally apply to other networks, e.g. parietal-prefrontal network, where persistent activity is seen during working memory tasks55,88. Indeed, recent studies of brain activity in humans has shown that functional network connectivity during spontaneous epochs is highly dynamic89, and that persistent activity during working memory gates the propagation of activity, and thus information, into the prefrontal network90.
In sum, these results demonstrate that during UDS, the rich dynamics of the entire cortico-entorhinal circuit can be captured in a quantitatively precise fashion by a dynamic attractor landscape involving just two biologically important variables: the cortico-entorhinal excitation and the recurrent excitation within the entorhinal cortex. Our model is simple enough to be analytically tractable. Despite the apparent simplicity and with just two parameters, we were able to reproduce nearly a dozen different experimental observations in a quantitatively precise fashion. This provides a strong support for our model to reveal the nature of cortico-entorhinal functional connectivity during slow oscillations in vivo, and the differential nature of this connectivity between MECIII vs LECIII. This approach provides a powerful technique to understand the functional connectivity between large networks of neurons in vivo.
Funding
This work was funded by the W. M. Keck Foundation, an AT&T research grant, National Science Foundation grant #1550678 and National Institutes of Health grant #1U01MH115746, all to MRM. Preliminary findings were presented in Society for Neuroscience meetings (2017-2019).
Author Contributions
KC did theory, simulations, and data analysis. SB and TGH did experiments. MRM participated and supervised all aspects. KC and MRM wrote the paper, with input from SB.
Competing Interests
The authors declare no competing interests.
Data and Materials Availability
Data and code are available upon reasonable request.
Supplementary Information
Materials and Methods
Supplementary Fig. 1 to Fig. 11
Acknowledgements
We thank J. McFarland for doing initial analysis, and noticing SPI, and J. Moore for comments on the manuscript. This work used computational and storage services associated with the Hoffman2 Shared Cluster provided by UCLA Institute for Digital Research and Education’s Research Technology Group.
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