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Wrote the paper: DB AW FW TG. Conceived and designed the models: DB FW TG. Contributed statistical tools: AW. Performed simulations: DB. Analyzed simulation results: DB AW.

The authors have declared that no competing interests exist.

Anatomic connections between brain areas affect information flow between neuronal circuits and the synchronization of neuronal activity. However, such structural connectivity does not coincide with effective connectivity (or, more precisely, causal connectivity), related to the elusive question “Which areas cause the present activity of which others?”. Effective connectivity is directed and depends flexibly on contexts and tasks. Here we show that dynamic effective connectivity can emerge from transitions in the collective organization of coherent neural activity. Integrating simulation and semi-analytic approaches, we study mesoscale network motifs of interacting cortical areas, modeled as large random networks of spiking neurons or as simple rate units. Through a causal analysis of time-series of model neural activity, we show that different dynamical states generated by a same structural connectivity motif correspond to distinct effective connectivity motifs. Such effective motifs can display a dominant directionality, due to spontaneous symmetry breaking and effective entrainment between local brain rhythms, although all connections in the considered structural motifs are reciprocal. We show then that transitions between effective connectivity configurations (like, for instance, reversal in the direction of inter-areal interactions) can be triggered reliably by brief perturbation inputs, properly timed with respect to an ongoing local oscillation, without the need for plastic synaptic changes. Finally, we analyze how the information encoded in spiking patterns of a local neuronal population is propagated across a fixed structural connectivity motif, demonstrating that changes in the active effective connectivity regulate both the efficiency and the directionality of information transfer. Previous studies stressed the role played by coherent oscillations in establishing efficient communication between distant areas. Going beyond these early proposals, we advance here that dynamic interactions between brain rhythms provide as well the basis for the self-organized control of this “communication-through-coherence”, making thus possible a fast “on-demand” reconfiguration of global information routing modalities.

The circuits of the brain must perform a daunting amount of functions. But how can “brain states” be flexibly controlled, given that anatomic inter-areal connections can be considered as fixed, on timescales relevant for behavior? We hypothesize that, thanks to the nonlinear interaction between brain rhythms, even a simple circuit involving few brain areas can originate a multitude of effective circuits, associated with alternative functions selectable “on demand”. A distinction is usually made between structural connectivity, which describes actual synaptic connections, and effective connectivity, quantifying, beyond correlation, directed inter-areal causal influences. In our study, we measure effective connectivity based on time-series of neural activity generated by model inter-areal circuits. We find that “causality follows dynamics”. We show indeed that different effective networks correspond to different dynamical states associated to a same structural network (in particular, different phase-locking patterns between local neuronal oscillations). We then find that “information follows causality” (and thus, again, dynamics). We demonstrate that different effective networks give rise to alternative modalities of information routing between brain areas wired together in a fixed structural network. In particular, we show that the self-organization of interacting “analog” rate oscillations control the flow of “digital-like” information encoded in complex spiking patterns.

In Arcimboldo's (1527–1593) paintings, whimsical portraits emerge out of arrangements of flowers and vegetables. Only directing attention to details, the illusion of seeing a face is suppressed (

This is illustrated by the example of a Giuseppe Arcimboldo's painting (

Brain functions –from vision

Several experimental and theoretical studies have suggested that

Ongoing local oscillatory activity modulates rhythmically neuronal excitability

Through large-scale simulations of networks of spiking neurons and rigorous analysis of mean-field rate models, we model the oscillatory dynamics of generic brain circuits involving a small number of interacting areas (

Through our analyses, we first confirm the intuition that “causality follows dynamics”. Indeed we show that our causal analysis based on TE is able to capture the complex multi-stable dynamics of the simulated neural activity. As a result, different

We show then that transitions can be reliably induced through brief transient perturbations properly timed with respect to the ongoing rhythms, due to the non-linear phase-response properties

Finally, we find that “information follows causality” (and, thus, again, dynamics). As a matter of fact, effective connectivity is measured in terms of time-series of “LFP-like” signals reflecting collective activity of population of neurons, while the information encoded in neuronal representations is carried by spiking activity. Therefore an effective connectivity analysis –even when based on TE– does not provide an actual description of information transmission in the sense of neural information processing and complementary analyses are required to investigate this aspect. Based on a general information theoretical perspective, which does not require specifying details of the used encoding

Our results provide thus novel theoretical support to the hypothesis that dynamic effective connectivity stems from the self-organization of brain rhythmic activity. Going beyond previous proposals, which stressed the importance of oscillations for feature binding

In order to model the neuronal activity of interacting areas, we use two different approaches, previously introduced in

A: in the network model, each local area is modeled as a large network of randomly and sparsely interconnected excitatory and inhibitory spiking neurons (inhibitory cells and synapses are in blue, excitatory cells and synapses are in red,

In a second analytically more tractable approach, each area is described by a mean-field firing rate variable (

For weak inter-areal coupling strengths, out-of-phase lockings of local periodic oscillations give rise to a family of “unidirectional driving” effective motif. The figure shows dynamics and corresponding effective connectivities for fully symmetric structural motifs with

The figure shows dynamics and corresponding effective connectivities for fully symmetric structural motifs with

The figure shows dynamics and corresponding effective connectivities for fully symmetric structural motifs with

For simplicity, we study fully connected structural motifs involving a few areas (

In the simple structural motifs we consider, delays and strengths of local excitation and inhibition are homogeneous across different areas. Long-range inter-areal connections are as well isotropic, i.e. strengths and delays of inter-areal interactions are the same in all directions. Delay and strength of local and long-range connections can be changed parametrically, but only in a matching way for homologous connections, in such a way that the overall topology of the structural motif is left unchanged. As previously shown in

We generate multivariate time-series of simulated “LFPs” in different dynamical states of our models and we calculate TEs for all the possible directed pairwise interactions. We show then that effective connectivities associated to different dynamical states are also different. The resulting effective connectivities can be depicted in diagrammatic form by drawing an arrow for each statistically significant causal interaction. The thickness of each arrow encodes the strength of the corresponding interaction. This graphical representation makes apparent, then, that effective connectivity motifs or, more briefly,

We analyze in detail, in

A first family of effective motifs occurs for weak inter-areal coupling. In this case, neuronal activity oscillates in a roughly periodic fashion (

A second family of effective motifs occurs for intermediate inter-areal coupling. In this case, the periodicity of the “LFP” oscillations is disrupted by the emergence of large correlated fluctuations in oscillation cycle amplitudes and durations. As a result, the phase-locking between “LFPs” becomes only approximate, even if it continues to be out-of-phase on average. The rhythm of the laggard area is now more irregular than the rhythm in the leader area. Laggard oscillation amplitudes and durations in fact fluctuate chaotically (

Finally, a third family of effective motifs occurs for stronger inter-areal coupling. In this case the rhythms of all the areas become equally irregular, characterized by an analogous level of fluctuations in cycle and duration amplitudes. During brief transients, leader areas can still be identified, but these transients do not lead to a stable dynamic behavior and different areas in the structural motif continually exchange their leadership role (

Further increases of the inter-areal coupling strength do not restore stable phase-locking relations and, consequently, do not lead to additional families of effective motifs. Note however that the effective motif families explored in

To conclude, we remark that absolute values of TE depend on specific parameter choices (notably, on time-lag and signal quantization, see

How can asymmetric causal influences emerge from a symmetric structural connectivity? A fundamental dynamical mechanism involved in this phenomenon is known as

The existence of a symmetry-breaking phase transition in the simple structural motifs we analyze here (for simplicity, we consider the

A: symmetric structural motifs can give rise to asymmetric dynamics in which one area leads in phase over the other (spontaneous symmetry breaking). Basins of attraction (in phase-shift space) of distinct phase-locking configurations are schematically shown here (for

Because of multi-stability, transitions between effective motifs within a family can be triggered by transient perturbations, without need for structural changes. We theoretically determine conditions for such transitions to occur. The application of a pulse of current of small intensity

In

A second non-linear dynamic mechanism underlying the sequence of effective motifs of

We consider as before a rate model of

A: examples of rate oscillations for different values of the inter-areal coupling in the rate model (

Bifurcation diagrams for the leader and for the laggard area are plotted in

For a sufficiently strong inter-areal coupling, symmetry in the dynamics of the bidirectional structural motif is suddenly restored

Despite its name, Transfer Entropy is not directly related to a transfer of information in the sense of neuronal information processing. The TE from area

In order to address this issue, we first introduce a framework in which to quantify the amount of information exchanged by interacting areas. In the case of our model,

A: in the case of sparsely synchronized oscillations, individual neurons fire irregularly (see four example spike trains, middle row) even when the local area activity undergoes a very regular collective rhythm (evident in “LFP” traces, bottom row). Therefore, a large amount of information can be potentially encoded, at every (analog) oscillation cycle, in the form of (digital-like) codewords in which “1” or “0” entries denote respectively firing or missed firing of a specific neuron in the considered cycle (top row). B: the strength of specific subsets of long-range excitatory synapses is systematically enhanced in order to form unidirectional “transmission lines” (TLs) embedded into the

We focus here on the fully symmetric structural motif of

The information transmission efficiency of each TL, for the case of different effective motifs, is then assessed by quantifying the Mutual Information (MI)

We have shown that a simple structural motif of interacting brain areas can give rise to multiple effective motifs with different directionality and strengths of effective connectivity, organized into different families. Such effective motifs correspond to distinct dynamical states of the underlying structural motif. Beyond this, dynamic multi-stability makes the controlled switching between effective motifs within a same family possible without the need for any structural change.

On the contrary, transitions between effective motifs belonging to different families (e.g. a transition from a unidirectional to a leaky driving motif) cannot take place without changes in the strength of the delay of inter-areal couplings, even if the overall topology of the underlying structural motif does not need to be modified. Each specific effective motif topology (i.e. motif family) is robust within broad ranges of synaptic conductances and latencies, however if parameters are set to be close to critical transition lines separating different dynamical regimes, transitions between different families might be triggered by moderate and unspecific parameter changes. This could be a potential role for neuromodulation, known to affect the net efficacy of excitatory transmission and whose effect on neural circuits can be modeled by coordinated changes in synaptic conductances

Note that dynamical coordination of inter-areal interactions based on precisely-timed synchronous inputs would be compatible with experimental evidence of phase-coding

In general, control protocols different from the one proposed here might be implemented in the brain. For instance, phased pulses might be used as well to stabilize effective connectivity in the presence of stronger noise. Interestingly, the time periods framed by cycles of an ongoing oscillation can be sliced into distinct functional windows in which the application of the same perturbation produces different effects.

Finally, in addition to “on demand” transitions, triggered by exogenous –sensory-driven– or endogenous –cognitive-driven– control signals, noise-driven switching between effective motifs might occur spontaneously, yielding complex patterns of activity during resting state

As revealed by our discussion of spontaneous symmetry breaking and effective entrainment, an analysis based on TE provides a description of complex inter-areal interactions compliant with a dynamical systems perspective. It provides, thus, an intuitive representation of dynamical states that is in the same “space” as anatomical connectivity.

Note that it is currently debated whether TE should be considered as a measure of effective connectivity in strict sense

TE constitutes thus a model-free approach (although, non “parameter-free”, cfr. forthcoming section and

A–C: The matrices in these panels illustrate the dependence of TE (network model,

Note that we do not intend to claim superiority of TE in some general sense. As a matter of fact TE is equivalent to GC, as far as the statistics of the considered signals are gaussian

We finally would like to stress, to avoid any potential confusion, that the structural motifs analyzed in the present study are well distinct from causal graphical models of neural activity, in the statistical sense proper of DCMs

The effective connectivity analyses presented in this study were conducted by evaluating TEs under specific parameter choices. However, absolute values of TE depend on parameters, like, notably, the resolution at which “LFP” signals are quantized and the time-lag at which we probe causal interactions. As discussed in detail in the

Considering the dependency on time-lag

Considering then the dependence on signal quantization, we observe that TE values tend to grow for increasing number of bins

By evaluating a threshold for statistical significance independently for each direction and combination of

We can thus summarize the previous statements by saying that absolute values of TE depend on the choices of

Traditionally, studies about communication-through-coherence or long-range binding between distant cell assemblies have emphasized the importance of in-phase locking (see, e.g.

In particular, our study confirms that the reorganization of oscillatory coherence might regulate the relative weight of bottom-up and top-down inter-areal influences

As a next step, we directly verified that “information follows causality”, since changes in effective connectivity are paralleled by reconfiguration of inter-areal communication modalities. Following

Therefore, by combining TE analyses of “LFP”-based effective connectivity with MI analyses of spike-based information transmission, we are able to establish a tight link between control of effective connectivity and control of communication-through-coherence, both of them being emergent manifestations of the self-organized dynamics of interacting brain rhythms.

To conclude, we also note that similar mechanisms might be used beyond the mesoscale level addressed here. Multi-stabilities of structural motifs might be preserved when such motifs are interlaced as modules of a network at the whole-brain level

The previous discussions suggest that oscillations, rather than playing a direct role in the representation of information, would be instrumental to the reconfigurable routing of information encoded in spiking activity. Original formulations of the communication-through-coherence hypothesis

As a matter of fact, our study reveals that the inherent advantages of “labelled-line” codes

This is particularly interesting, since many cortical rhythms are only sparsely synchronized, with synchronous oscillations evident in LFP, Multi-Unit Activity or intracellular recordings but not in single unit spike trains

It is very plausible that flexible inter-areal coordination is achieved in the brain through dynamic self-organization

Another open question is whether our theory can be extended to encompass the control of effective connectivity across multiple frequency bands

Finally, we are confident that our theory might inspire novel experiments, attempting to manipulate the directionality of inter-areal influences via local stimulation applied conditionally to the phase of ongoing brain rhythms. Precisely timed perturbing inputs could indeed potentially be applied using techniques like electric

Each area is represented by a random network of

The membrane potential is given by:

Short-range connections within a local area

First, a structural motif of interconnected random networks of spiking neurons is generated, as in the previous section. Then, on top of the existing excitatory long-range connections, additional stronger long-range connections are introduced in order to form directed transmission lines. In each area a source sub-population, made out of 400 excitatory neurons, and a non-overlapping target sub-population, made out of 200 excitatory and 200 inhibitory neurons, are selected randomly. Excitatory cells in the source populations get connected to cells in the target sub-populations of the other area via strong synapses. These connections are established in a one-to-one arrangement (e.g. each source cell establishes a TL-synapse with a single target cell that does not receive on its turn any other TL-synapse).

The peak conductance

Each area is represented by a single rate unit. The dynamical equations for the evolution of the average firing rate

Given an oscillatory time-series of neuronal activity, generated indifferently by a rate or by a network model, a phase

The phase shift induced by a pulse perturbation

In the network model, it is possible to evaluate the phase-shift induced by a perturbation, by directly simulating the effects of this perturbation on the oscillatory dynamics. A perturbation consists of a pulse current of strength

For simplicity, we focus in the following on the case of

For the network model, the waveform of “LFP” oscillations can be determined through simulations. Since not all oscillation cycles are identical, the limit cycle waveform is averaged over 100 different cycles –as for the determination of

Phase intervals in which the application of a pulsed perturbation leads to a change of effective connectivity directionality are determined theoretically as shown below. For

Let us consider first a structural motif with

This quantity represents the Kullback-Leibler divergence

We also measure the causal unbalancing

Considering now a structural motif with

Absolute values of

When considering a structural motif involving more areas, the “LFP” time-series of each area can be resampled jointly or independently. When resampling jointly, matching starting points and block-lengths are selected for each block of the resampled time-series of each area, leading to resampled multivariate time-series in which the structure of causal influences should not be altered. The distribution of

Note that geometric bootstrap can be applied to arbitrary signals, and does not depend on their strict periodicity. However it is precisely the strong periodic component of our signals that makes necessary the use of geometric bootstrap techniques. Indeed, conventional bootstrap, strongly disrupting signal periodicity, would lead to artificially low thresholds for statistical significance of TE (not shown).

We evaluate information transmission between pairs of mono-synaptically connected cells in different areas, linked by a TL-synapse (TL pairs) or by a normally weak long-range synapse (control pairs). Inspired by

Statistics are taken over 400 pairs of cells per synapse set, i.e. one set of strong synapses per embedded TL, plus one set of (control) weak synapses. The box-plots in

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We thank Albert Compte for useful suggestions and Olav Stetter for figure editing. DB wishes to credit Nicolas Brunel and David Hansel for earlier collaboration on the models used in this study.