A Control-Theoretic Formulation of Global Workspace Theory

Abstract: Global workspace theory explains conscious access as the broadcasting of selected information to the rest of the network, but it lacks a formal criterion for identifying the mechanism that enables this access. We propose that a global workspace is a mediator, namely, a subnetwork that receives activity from distributed systems, transforms it through internal modes, and returns differentiated effects to the broader network. We formalize this claim as the Global Mediation Workspace (GMW), a control-theoretic formulation in which a candidate subnetwork is treated as an open system embedded in the remainder of the network. In this framework, reachability characterizes how the remainder can drive the candidate, observability characterizes how candidate states affect the remainder, and a boundary Hankel operator identifies the internal modes linking the two directions. The resulting signature quantifies mediation capacity, input-output alignment, effective dimensionality, and routed source-target breadth, each of which characterizes different components of global workspace. In synthetic benchmarks, we tested whether the signature can distinguish a planted differentiated mediator from dense hubs, one-sided receivers or broadcasters, and a split read/write aggregate with no common internal route. A nonlinear extension characterizes mediation through trajectory-conditioned differential operators, finite-amplitude response profiles, and state-dependent coalitions. As a preliminary application, we estimated the signature from ECoG recordings in four macaques under ketamine anesthesia. We found that input-output alignment was reduced during unconsciousness whereas potential capacity was increased. The GMW thus provides a formal and testable criterion for locating candidate global workspaces in neural recordings and for asking which aspects of mediation is crucial for conscious access.

• The paper formulates the Global Mediation Workspace as an open subsystem whose boundary Hankel operator identifies internal modes that are both reachable from and observable in the surrounding network.
• Synthetic benchmarks show that the proposed Workspace Mediation Index ranked the planted mediator first, separating it from split input/output decoys, one-sided nodes, peripheral sets, and nearly rank-one hubs.
• Macaque ECoG results show that deep anesthesia increased predictability and mediation gain but reduced alignment, differentiated organization, and routed breadth, demonstrating why a single scalar score cannot establish conscious access.

Global workspace theory characterizes conscious access as the availability of selected information to multiple specialized systems, but its central organizational mechanism has remained difficult to identify operationally. “A Control-Theoretic Formulation of Global Workspace Theory” (2608.15926) addresses this problem by defining the Global Mediation Workspace (GMW): a candidate subnetwork that receives activity from the rest of the system, transforms it through internal dynamical modes, and returns differentiated effects to the same external network. The proposal is explicitly relational and dynamical. It does not identify consciousness with a fixed anatomical hub, a scalar network statistic, or any single neural signature.

From global availability to boundary mediation

The paper’s central conceptual move is to treat a candidate subnetwork $S$ as an open system embedded in a remainder $R=V\setminus S$. The candidate has internal dynamics, receives boundary inputs from $R$, and produces boundary outputs that influence $R$:

$z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$

Here, $A_S$ describes candidate-internal dynamics, $B_S$ maps activity from the remainder into the candidate, and $C_S$ maps candidate states back to the remainder. The same specialist systems can therefore act as sources, targets, or both. This is important because the proposed workspace operation is not a fixed feedforward chain but a many-to-many transformation of the form $R\rightarrow S\rightarrow R$.

The formulation distinguishes several structures that can appear globally connected while failing to implement this operation. A dense hub may have high degree but transmit only one redundant mode. A receiver may encode external activity without influencing the rest of the network, while a broadcaster may exert broad effects without being meaningfully driven. A split read/write aggregate may contain strong receivers and broadcasters but lack any internal route connecting the two. The GMW is intended to reject all of these cases.

The paper formalizes the candidate boundary using finite-horizon reachability and observability. Past boundary inputs are mapped into candidate states by a reachability matrix, while candidate states are mapped into future boundary outputs by an observability matrix. Their product is a finite-horizon boundary Hankel operator:

$H_L(S)=O_L(S)R_L(S).$

Its blocks are $R=V\setminus S$0, which aggregate signed, gain-weighted paths that enter the candidate, propagate internally, and return to the boundary. The nonzero singular values of this operator identify internal directions that are jointly reachable from and observable through the remainder.

Figure 1: The GMW treats a candidate as an open subsystem whose boundary Hankel operator identifies modes jointly reachable from and observable in the remainder.

This construction provides a stricter criterion than bidirectional connectivity. A path must enter the candidate, use its internal dynamics, and produce an external consequence. Direct paths that bypass $R=V\setminus S$1 are excluded. In observational neural recordings, however, the input term represents modeled activity in the remainder rather than independently manipulated interventions. Accordingly, the resulting Gramian should be interpreted as boundary reachability under a fitted dynamical model, not as proof that the candidate can be experimentally controlled through arbitrary external inputs.

Capacity, alignment, dimensionality, and routing

A principal contribution is the decomposition of mediation into distinct components rather than the use of a single undifferentiated score. The reachability and observability Gramians characterize receive-side and send-side capacity separately. Their eigenvectors specify the internal state directions that can be driven and expressed, respectively. Mediation depends on whether these directions align.

The paper defines a spectrum-matched capacity envelope, denoted $R=V\setminus S$2, representing the maximum mediation compatible with the separate reachability and observability spectra under optimal mode pairing. The realized mediation strength is

$R=V\setminus S$3

where the nuclear norm sums the boundary-Hankel singular values. Alignment efficiency is then

$R=V\setminus S$4

This distinction gives the framework a useful diagnostic interpretation. A candidate can possess substantial potential capacity while realizing little of it because its receive and send modes are poorly aligned. Conversely, a candidate can exhibit near-perfect alignment but remain nearly rank one, meaning that most external interactions are mediated through a single bottleneck.

The mediation singular-value distribution supplies an entropy effective rank, $R=V\setminus S$5, which measures the number of substantial internal mediation modes. The paper also introduces $R=V\setminus S$6, a routed source–target breadth measure. Rather than counting incident edges or module contacts, it computes how mediation energy is distributed across ordered pairs of specialist modules. The resulting four-component signature is:

$R=V\setminus S$7

The authors emphasize that no single component is stipulated to define consciousness. The signature separates potential capacity, realized alignment, differentiated dimensionality, and many-to-many routing. This is theoretically consequential: a high mediation magnitude can coexist with a low-dimensional or poorly routed organization, and therefore cannot by itself establish a workspace-like mechanism.

For fixed-size candidate search, the paper uses the Workspace Mediation Index:

$R=V\setminus S$8

WMI is explicitly pragmatic rather than foundational. Its multiplicative form penalizes candidates that fail strongly on any component, but the equal weighting implicit in the product is not uniquely justified. Candidate size, horizon, state metric, module partition, and minimum internal path length must all be declared in advance. The paper also stresses that WMI does not determine a canonical or minimal workspace: adding a useful node can continue to increase capacity or routing breadth.

Synthetic validation against structurally misleading candidates

The synthetic benchmark provides the strongest causal validation because the generating network and planted mediator are known. It contains 64 directed nodes: four 12-node specialist modules, a planted four-node GMW, actuator-only nodes, observer-only nodes, a split input/output decoy, and a dense approximately rank-one hub. The full coupling matrix was scaled to spectral radius $R=V\setminus S$9, with primary horizon $R$0.

Figure 2: The synthetic benchmark embeds a planted four-node mediator alongside one-sided, split, peripheral, and dense low-rank decoys.

The successive criteria demonstrate why the boundary-Hankel construction is necessary. A naive external-access measure assigned the split input/output decoy a score of $R$1, exceeding the planted GMW’s $R$2. Thus, separate receive and send access favored a false positive. Requiring internal boundary mediation reduced the split decoy’s WMI to $R$3, while the planted GMW scored $R$4. Replacing direct-edge module breadth with routed source–target breadth further reduced a peripheral false positive to $R$5.

Figure 3: Internal mediation rejects a split receiver–broadcaster aggregate, while routed breadth rejects peripheral sets with broad incident connectivity but narrow mediated transformations.

The capacity–alignment decomposition explains these results quantitatively. The split decoy had a substantial potential capacity envelope of $R$6, but realized mediation was only $R$7, corresponding to alignment efficiency $R$8. Its two dominant principal-angle cosines were only $R$9 and $R$0, indicating nearly orthogonal receive and send subspaces. The decoy therefore had the machinery to receive and broadcast activity separately, but not to mediate the same internal states between them.

The dense hub produced the opposite failure mode. Its capacity envelope and realized mediation were almost identical: $R$1, $R$2, and $R$3. However, its effective rank was only $R$4. The hub was therefore strongly aligned but almost entirely one-dimensional. By contrast, the planted GMW combined $R$5, $R$6, $R$7, effective rank $R$8, and routed breadth $R$9.

Candidate	$z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$0	$z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$1	$z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$2	$z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$3	$z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$4	WMI
Planted GMW	3.699	0.978	3.616	3.839	0.934	3.242
Dense degree hub	2.882	1.000	2.881	1.002	0.926	0.668
Split I/O decoy	2.775	0.145	0.401	2.961	0.825	0.245
Actuator-only set	0.146	0.899	0.131	3.399	0.701	0.078
Observer-only set	0.112	0.965	0.108	3.477	0.860	0.080
Random peripheral set	0.771	0.869	0.670	3.731	0.182	0.114

Figure 4: The decoys fail for distinct reasons: poor alignment in the split set, low capacity in one-sided sets, and rank collapse in the dense hub.

The spectral structure reinforces the distinction. The dense hub’s mediation spectrum was approximately $z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$5, whereas the GMW contained four substantial modes. Moreover, the GMW distributed mediated energy across all 16 ordered specialist-module pairs. The hub contacted many module pairs, but largely through one internal mode. This result directly supports the paper’s claim that broad connectivity and broad differentiated mediation are not equivalent.

Figure 5: The planted GMW supports several substantial mediation modes and distributes energy across source–target module pairs, unlike the nearly rank-one dense hub.

The exhaustive search over all $z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$6 four-node subsets ranked the planted GMW first by WMI, with score $z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$7. The second-ranked set, $z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$8, scored $z_{t+1}=A_S z_t+B_Su_t,\qquad y_t=C_Sz_t.$9, and all ten highest-scoring sets contained at least three planted GMW nodes. In 50 independently generated networks, width-50 beam search recovered all four planted nodes in 45 cases, with mean Jaccard overlap $A_S$0.

These results establish construct and numerical validity within the declared synthetic regime. They do not establish biological validity, and the favorable performance partly reflects a benchmark deliberately designed around the proposed confounds. The comparison with conventional measures is consequently informative but not decisive evidence that WMI is superior in natural neural networks. For example, betweenness also ranked the planted GMW first in this particular network, although it cannot quantify mode alignment or mediation dimensionality.

Figure 6: Conventional metrics favor different decoys, whereas WMI ranks the planted GMW first in the exhaustive fixed-size search.

Nonlinear mediation and context-dependent coalitions

The linear formulation is exact only for a declared linear operator. In nonlinear systems, mediation depends on the reference trajectory, perturbation amplitude, input ensemble, and operating regime. The paper therefore extends the construction in two complementary directions.

First, the Jacobians along a reference trajectory define a trajectory-conditioned differential boundary operator. Its singular spectrum characterizes local mediation around that trajectory. This is a local, infinitesimal quantity and may fail to detect nonlinear gates that open only at finite amplitude. Second, the authors define a secant operator over perturbations of amplitude $A_S$1, allowing the mediation spectrum to be evaluated for finite-amplitude responses.

The nonlinear simulations demonstrate that these distinctions matter. In a tanh benchmark, moderate gain preserved recovery of the planted mediator, whereas sufficiently high gain changed the active functional operator and caused even oracle operators to favor another candidate. Passive VAR estimation was strongly sample dependent: at tanh gain $A_S$2, increasing observed transitions from 500 to 32,000 improved the planted candidate’s rank from 36,319 to 1. This is an estimation effect rather than evidence that passive dynamics are intrinsically inadequate. In a ReLU benchmark at bias $A_S$3, the central secant ranked the planted candidate first, the mean Jacobian second, and passive VAR 245th. At bias $A_S$4, all three ranked it first.

These results imply that recovery claims must distinguish structural recovery, functional recovery, and statistical recovery. A passive estimator can fail because of finite samples even when the underlying operator is unchanged; conversely, an oracle estimator can fail because nonlinear operating conditions have genuinely altered the mediated route.

The finite-amplitude example is particularly relevant to the paper’s interpretation of ignition. The secant mediation strength peaked near input amplitude $A_S$5, with $A_S$6 and effective rank approximately $A_S$7, then declined to $A_S$8 at amplitude 2 because of saturation. Thus, mediation is not necessarily monotonic in perturbation amplitude. A threshold-like rise in the GMW signature can coexist with saturation at larger amplitudes.

Figure 7: Finite-amplitude mediation increases when a gated route opens and declines when the route saturates.

A bistable context simulation further produced hysteresis: opening and closing transitions occurred at drives $A_S$9 and $B_S$0, and at zero drive the open and closed branches had WMI values of $B_S$1 and $B_S$2, respectively. This demonstrates that the same nominal external drive can correspond to different mediation states depending on trajectory history. It is a property of the simulated bistable system, not evidence of pharmacological hysteresis in the macaque data.

The nonlinear formulation also permits state-dependent candidate selection. The currently instantiating coalition is defined as the fixed-size candidate maximizing differential WMI under the current trajectory. In simulations, the maximizing coalition shifted from visual to multimodal to auditory candidates as sensory context changed, despite an unchanged anatomical network.

Figure 8: State-dependent recruitment changes the coalition currently implementing the GMW without requiring anatomical relocation.

This result supports a distinction between anatomical availability and functional instantiation. A route may exist structurally but remain inactive because the operating state does not align its receive and send modes. Conversely, a fixed anatomical substrate can recruit different node coalitions under different contexts.

Macaque ECoG application

The empirical analysis uses geometry-audited, nonoverlapping bipolar ECoG recordings from four macaques across 11 ketamine–medetomidine experiments. Candidate sizes $B_S$3 were prespecified. Dynamics were fitted at a 25-ms lag, with $B_S$4 and internal shift $B_S$5, corresponding to input–output separations from 50 to 200 ms. Candidate discovery and state evaluation were separated using within-day cross-fitting and same-animal leave-one-day-out transfer.

The authors explicitly frame this as a preliminary application, not a test of GWT. The aim is to determine whether the signature can be estimated from real recordings and whether its components separate during anesthesia. This distinction is methodologically appropriate because the ECoG analysis does not independently establish conscious content or causal necessity.

Deep anesthesia produced a striking dissociation between dynamical gain and organization. Held-out short-lag prediction increased on every experiment day, with median $B_S$6 rising from $B_S$7 in wakefulness to $B_S$8 during deep anesthesia. At candidate size $B_S$9, the animal-balanced deep/awake ratios were:

• $C_S$0: $C_S$1 with descriptive interval $C_S$2;
• $C_S$3: $C_S$4 with interval $C_S$5;
• $C_S$6: $C_S$7 with interval $C_S$8.

Thus, anesthesia increased realized mediation and potential capacity while reducing alignment. This is a directly contradictory pattern relative to any interpretation of high predictability or high mediation magnitude as sufficient evidence of differentiated global organization.

The reductions in organization were clearest for $C_S$9. At that size, the deep/awake ratios were $R\rightarrow S\rightarrow R$0 for alignment, $R\rightarrow S\rightarrow R$1 for effective-rank fraction, $R\rightarrow S\rightarrow R$2 for routed breadth, and $R\rightarrow S\rightarrow R$3 for gain-free organization; top-mode share increased to $R\rightarrow S\rightarrow R$4. Raw WMI nevertheless remained above one at every candidate size: $R\rightarrow S\rightarrow R$5, $R\rightarrow S\rightarrow R$6, and $R\rightarrow S\rightarrow R$7 for $R\rightarrow S\rightarrow R$8. The implication is central to the paper’s empirical argument: a scalar WMI can increase when gain rises enough to offset losses in alignment, dimensionality, and routing. Full signature reporting is therefore necessary.

The gain increase survived several controls, including state-wise variance normalization, restriction above 4 Hz, delta-only restriction, and removal of the leading state-specific principal component. However, the authors acknowledge that these controls do not replace phase-randomized or autocorrelation-matched surrogates. Slow temporal structure, spectral concentration, and the fitted observation model remain possible contributors.

Figure 9: Realized mediation strength remains elevated during deep anesthesia under several slow-wave and variance controls.

Figure 10: Gain-free organization remains reduced under the same controls, particularly for larger candidates and frequencies above 4 Hz.

Candidate sites selected repeatedly in awake data were broadly distributed across frontal, premotor, parietal, sensorimotor, temporal, and other association-related sectors. This is compatible with a distributed workspace scaffold rather than a unique focal locus, but the result is limited by incomplete cortical coverage, animal-specific electrode layouts, and the use of a common two-dimensional display that is not stereotactically registered.

Figure 11: Recurrently selected awake candidate sites span multiple cortical sectors; selection frequency is not a probability of causal necessity.

The corrected montage analysis is equally important. In two animals requiring geometry-aware rematching, the increase in gain and general reduction in alignment survived the change in montage, but exact component magnitudes and candidate identities remained montage dependent. For example, one animal’s $R\rightarrow S\rightarrow R$9 $H_L(S)=O_L(S)R_L(S).$0 ratio changed from $H_L(S)=O_L(S)R_L(S).$1 to $H_L(S)=O_L(S)R_L(S).$21.61$H_L(S)=O_L(S)R_L(S).$31.34. The robust inference is therefore the qualitative gain–alignment dissociation, not invariance of individual candidate sets.

Recovery was heterogeneous and cannot be interpreted as pharmacological hysteresis because anesthetic concentration was not matched between induction and recovery. Across nine confirmed recovery days from three animals, $H_L(S)=O_L(S)R_L(S).$4 remained at approximately $H_L(S)=O_L(S)R_L(S).$5–$H_L(S)=O_L(S)R_L(S).$6 of awake levels during eyes-closed recovery and $H_L(S)=O_L(S)R_L(S).$7–$H_L(S)=O_L(S)R_L(S).$8 during eyes-open recovery. Alignment approached baseline more closely at larger candidate sizes, while effective-rank fraction remained modestly reduced. These trajectories show that the signature’s components need not recover synchronously.

Limitations and open questions

The principal limitation is inferential scope. The GMW measures extrinsic mediation across a declared boundary; it does not measure intrinsic cause–effect irreducibility, phenomenal consciousness, or predictive autonomy. It therefore complements rather than replaces IIT, information-closure approaches, causal density, and information-decomposition methods. The paper does not claim that a high GMW signature is sufficient for consciousness.

The framework is also conditional on analyst-specified choices: candidate size, temporal horizon, internal shift, state metric, module partition, normalization, and scalarization. Nonorthogonal changes of internal coordinates preserve the total boundary operator but can alter the capacity–alignment decomposition unless the state metric is transformed consistently. The module-pair breadth term is likewise partition dependent. A canonical rule for choosing the boundary, state variables, spatial grain, and temporal grain remains absent.

The empirical operators are fitted predictive dynamics rather than direct synaptic causal models. Bipolar rereferencing, hidden sources, filtering, volume conduction, regularization, incomplete electrode coverage, and slow-wave structure all affect the estimated state space. The no-contact-reuse montage reduces algebraic redundancy but does not eliminate observation-model dependence. Exact candidate identity was less stable than the direction of state effects, with mean pairwise Jaccard overlap ranging from $H_L(S)=O_L(S)R_L(S).$9 to $R=V\setminus S$00 across animals and candidate sizes.

The nonlinear results also leave open how to choose between infinitesimal, finite-amplitude, trajectory-conditioned, and higher-order operators in biological data. A hard-threshold system can have an uninformative pathwise Jacobian while exhibiting informative ensemble-averaged switching. Conversely, finite-amplitude profiles depend on perturbation distributions and reference trajectories. These are not merely implementation details; they determine what counts as an active mediation route.

Finally, the paper does not determine whether conscious access depends primarily on capacity, alignment, effective rank, routed breadth, or a conjunction of these quantities. The proposed experiments—masking, attentional blink, no-report paradigms, perturbational assays, and state transitions—would need to specify candidates and output channels independently of the observed results. A central open question is whether phenomenal reports or other independently validated markers of conscious content covary with $R=V\setminus S$01, $R=V\setminus S$02, differentiated rank, and routed breadth, or whether these components dissociate systematically.

Conclusion

The paper provides a formal control-theoretic account of a workspace-like subnetwork as an internally mediating open system. Its boundary Hankel operator identifies modes that are simultaneously reachable from and observable in the network remainder, while the proposed signature separates potential capacity, mode alignment, differentiated dimensionality, and routed source–target breadth. Synthetic benchmarks show that these quantities distinguish a planted mediator from split, one-sided, peripheral, and low-rank hub decoys. Nonlinear analyses extend the framework to trajectory-dependent and finite-amplitude mediation. In macaque ECoG, anesthesia increased predictability and mediation gain but reduced alignment and, at larger candidate sizes, differentiated organization. The framework therefore supports a technically precise distinction between strong dynamics and globally differentiated mediation, while leaving the relation between these quantities and conscious access as an empirical question (2608.15926).