What Consciousness Requires: A Measurable, Physical Perspective

2. PHYSIOLOGICAL DYNAMICS UNDERLYING SHIFTS IN CONSCIOUSNESS

We examine here how key biophysical parameters behave across different states of consciousness, including wakefulness, sleep, anesthesia and coma. We synthesize empirical findings on oscillatory activity, energy metabolism, ionic regulation, thermodynamics, network connectivity and systemic physiological rhythms, highlighting their state-dependent variability. Each parameter is analyzed in terms of its measurable shifts and threshold behavior during transitions into and out of consciousness (Table 1). A qualitative visualization of how these multiple interdependent parameters tend to degrade progressively as consciousness diminishes is illustrated in Figure 1.

Oscillatory signatures across conscious states. Neural oscillations are among the most consistent indicators of conscious state transitions. In healthy wakefulness, cortical networks exhibit robust gamma-band activity (30–80 Hz), particularly within frontoparietal circuits (Mikulan et al., 2018). This coherence often decreases progressively during sleep stages and is further suppressed under anesthesia and coma (Mashour 2024). Through multiple studies, gamma power $P_{\gamma }\left(t\right)$ can be modeled as a decaying function of arousal index α, where:

$$P_{\gamma }\left(t\right)=P_0·e^{-k\alpha \left(t\right)}$$

Here, $P_0$ is the baseline power during conscious wakefulness and k is a state-dependent decay constant. In parallel, phase synchrony between brain regions, quantified by phase-locking values (PLVs), is found to collapse during transitions into non-REM sleep or anesthetic-induced unconsciousness (Zhou et al., 2020a; Leguia et al., 2021; Bardon et al., 2023). The decline in PLV values tends to precede behavioral loss of responsiveness, suggesting that coherent oscillatory activity is not only correlated with consciousness but may serve as a prerequisite. Variability in synchronization is often region-specific; posterior cortical areas retain partial coherence during light sedation, while prefrontal regions disconnect more fully (Liu et al., 2017; Zelmann et al., 2023). These patterns reinforce the notion that high-frequency oscillations and their coherence represent one of the earliest and most sensitive indicators of conscious degradation.

Energy metabolism and the consciousness spectrum. The metabolic demands of consciousness are among the most thoroughly characterized physiological dependencies in neuroscience (Engl and Attwell, 2015; Liu and Prokosch, 2021). Functional neuroimaging shows that conscious wakefulness is associated with elevated cerebral metabolic rates of glucose (CMRglu) and oxygen (CMRO₂), especially in cortical and thalamic regions (Bernier et al., 2017; We et al., 2022; Stankeviciute et al., 2023; Caporale et al., 2023). Quantitatively, the resting-state CMRglu in the awake human brain is approximately 5.5–6.0 mg/100g/min (Maquet et al., 1990). This value drops by 25–35% during non-REM sleep and can decrease by over 50% during deep anesthesia (Liu 2013). Similar patterns are found for CMRO₂, with reductions well below 2.5 mL/100g/min marking the transition into unconsciousness. These data can be represented as sigmoidal functions:

$$\mathrm{C}\mathrm{M}\mathrm{R}(s)={\displaystyle \frac{R_{\mathrm{m}\mathrm{a}\mathrm{x}}}{1+e^{-k(s-s_0)}}}$$

Here, s is the state index (e.g., 0 for coma, 1 for wake), R_max is the maximal rate in wakefulness and $s_0$ is the midpoint state where the rate is half-maximal. These curves can be used to approximate metabolic thresholds across consciousness conditions. Notably, these reductions are not globally uniform (Koush et al., 2021). Default mode network hubs like the medial prefrontal cortex and posterior cingulate cortex display disproportionate metabolic suppression during unconscious states (Liu et al., 2019). These regional vulnerabilities suggest that sustained consciousness requires not only high total metabolic input, but also its organized regional distribution. The empirical patterns confirm that metabolism operates as a limiting substrate, providing an energetic envelope necessary for neural integration and oscillatory coherence.

Ionic and synaptic stability under fluctuating states of consciousness. During conscious states, extracellular concentrations of potassium, sodium and calcium are tightly regulated by astrocytic uptake, active transport and blood-brain barrier dynamics (Sajib et al., 2018; DiNuzzo 2019; Cheng et al., 2024; Theparambil et al., 2020). Potassium concentration in the extracellular space typically ranges from 3.0 to 5.0 mM (Wellbourne-Wood et al., 2017; Walch et al., 2020). Elevations beyond this—often above 7.5–8.0 mM—are observed in hypoxic, epileptic or anesthetic conditions, correlating with decreased excitability due to depolarization block (Palmer and Clegg, 2017; Lindner et al., 2020). These shifts are modeled using dynamic Nernst potentials, where the membrane voltage is acutely sensitive to ionic drift (Song et al., 2018):

$$\mathsf{\Delta }{V}_m={\displaystyle \frac{RT}{zF}}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{[K^{+}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{n}\mathrm{e}\mathrm{w}}}{[K^{+}{]}_{\mathrm{o}\mathrm{u}\mathrm{t}}^{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}}}\right)$$

These changes result in altered excitability thresholds, leading to either spurious hyperexcitability or transmission failure. Similarly, the intracellular Ca²⁺ concentration, which supports synaptic vesicle release and plasticity, decreases in anesthetized and deeply sedated states, reducing the likelihood of coordinated neurotransmission (Leitz and Kavalali, 2016; Eisner et al., 2023). These ion fluxes are tightly tied to local metabolic conditions and ATP-dependent pumps such as the Na⁺/K⁺ ATPase consume substantial cellular energy to maintain these gradients. Overall, the breakdown of ionic gradients and the degradation of synaptic precision represent convergent, quantifiable disruptions that underpin many states of unconsciousness.

Temperature regulation and thermodynamic thresholds. Empirical studies of hypothermia, fever and therapeutic cooling demonstrate that even modest deviations from normothermia (~37°C) lead to significant impairments in consciousness and cognition. Temperature effects on neuronal firing can be modeled using the above-mentioned Arrhenius relation, where the rate constantfor enzymatic and synaptic processes depends exponentially on inverse temperature. Lower temperatures reduce the kinetics of ion channel opening, neurotransmitter release and mitochondrial ATP production. The resulting delays in synaptic transmission and integration times disrupt the fast feedback loops needed for high-frequency synchrony and global integration (Burek et al., 2019; Jabbari and Karamati, 2022). In states of hyperthermia (>41°C), excessive kinetic activity can cause denaturation of enzymes and failure of glial buffering systems. Thermoregulation also interacts with cerebral blood flow and oxygen availability; cold temperatures reduce perfusion, while excessive heat accelerates oxidative stress. These dynamics, modeled via the Pennes bioheat equation (Tucci et al., 2021), show how thermodynamic shifts can create physiological conditions incompatible with sustained consciousness. Temperature therefore constitutes a boundary parameter, defining the energetic feasibility space for cognitive activity.

Structural and functional network dynamics. Consciousness is increasingly understood as a network-level phenomenon, requiring both high integration and modular segregation of cortical and subcortical regions (Gonzalez-Escamilla et al., 2023; Wu et al., 2023; Nestor et al., 2024). Structural connectivity derived from diffusion tensor imaging (DTI) defines the anatomical scaffold over which functional interactions occur. In conscious individuals, these networks exhibit small-world properties: short path lengths and high clustering. These features are quantified via graph-theoretical metrics such as the above-mentiones global efficiency, path length L and modularity. Empirical findings show that unconscious states, whether induced pharmacologically or by trauma (Pullon et al., 2022; Choe et al., 2025), are associated with increased path length and reduced efficiency. Resting-state fMRI data confirms that connectivity within the default mode network (DMN) and between DMN and frontoparietal networks becomes attenuated under unconscious conditions (Long et al., 2016; Lemaire et al., 2022). Notably, consciousness seems to depend on the capacity of networks to maintain reciprocal communication— “reentry”—across distant cortical zones. This dynamic coupling is highly dependent on both structural links and phase-coherent oscillations, reinforcing the idea that consciousness is not reducible to local activity alone. Instead, it may reflect a system-wide capability for efficient information transfer across a physically defined architecture.

Overall, by mapping all the above-mentioned changes across physiological states, this chapter established the dynamic nature of consciousness as a function of coordinated, physically constrained variables.

3. CONVERGENCE OF METHODS IN THE PHYSICAL STUDY OF CONSCIOUSNESS

We explore here the convergence of evidence from multiple measurement modalities—such as EEG, fMRI, PET, DTI and physiological monitoring—in identifying the physical correlates of consciousness. We show that structurally and functionally distinct techniques yield overlapping signatures of conscious states, reinforcing the reliability of key parameters like gamma coherence, metabolic rate and network connectivity.

Convergence between electrophysiological and hemodynamic measures. One of the most robust intersections in consciousness research lies in the correlation between electrophysiological recordings (e.g., EEG, MEG) and hemodynamic imaging techniques (e.g., fMRI, PET) (Hall et al., 2014; Gomez-Pilar et al., 2022). Despite their distinct temporal and spatial resolutions, both modalities consistently identify overlapping brain regions and patterns associated with conscious awareness. For instance, studies combining EEG and fMRI during transitions between wakefulness, sleep and anesthesia frequently observe that decreases in gamma-band oscillatory power (30–80 Hz) are paralleled by reductions in BOLD signal amplitude, particularly in frontoparietal and default mode network regions (Yang et al., 2022; Guo et al., 2023; Vijayakrishnan eta l., 2023). These correlations can be quantified using time-lagged canonical correlation analysis (CCA) (Schultze and Grubmüller, 2021), which maps signal covariance structures across modalities:

$$\mathrm{C}\mathrm{C}\mathrm{A}:{\max }_{\overrightarrow{a},\overrightarrow{b}}{\displaystyle \frac{{\overrightarrow{a}}^T{\mathbf{X}}^T\mathbf{Y}\overrightarrow{b}}{\sqrt{{\overrightarrow{a}}^T{\mathbf{X}}^T\mathbf{X}\overrightarrow{a}·{\overrightarrow{b}}^T{\mathbf{Y}}^T\mathbf{Y}\overrightarrow{b}}}}$$

Here, X and Y are the EEG and fMRI data matrices, respectively. Strong canonical correlations are found particularly when EEG coherence in the gamma band is high and BOLD activation is prominent in higher-order associative cortices (Rodu et al., 2018). These findings indicate that the oscillatory synchrony needed for conscious integration has a vascular metabolic signature, validating both modalities as independent but convergent proxies for the conscious state.

Correlating metabolic demand with electrophysiological dynamics. A critical relationship does exist between cerebral energy metabolism and electrophysiological activity, particularly in the high-frequency gamma band. Metabolic imaging techniques such as FDG-PET and calibrated fMRI offer quantitative estimates of glucose and oxygen consumption, which are then correlated with EEG-derived measures of neural activity. In healthy wakeful subjects, increased gamma coherence in the cortex correlates strongly with higher regional cerebral metabolic rates of glucose (CMRglu). This relationship follows a near-linear trend under normal conditions, as neuronal firing frequency is energetically expensive and gamma-band oscillations require high synchrony and rapid synaptic turnover. The dependence of electrical activity on metabolic support is mathematically formalized in energy-flow coupling models:

$$P_{\mathrm{E}\mathrm{E}\mathrm{G}}\left(f\right)\propto \mathrm{C}\mathrm{M}\mathrm{R}\mathrm{g}\mathrm{l}\mathrm{u}·{\displaystyle \frac{1}{\tau \left(f\right)}}$$

Here, $\tau \left(f\right)$ is the frequency-dependent time constant for synaptic recovery and ATP replenishment. As the metabolic substrate becomes limited (e.g., under anesthesia or ischemia), high-frequency oscillatory power drops, particularly in associative cortical areas. Conversely, the restoration of metabolic support is necessary for the reappearance of gamma activity during recovery from unconsciousness. The spatial topography of metabolic and electrical changes is consistent, with posterior and medial hubs being particularly energy dependent. This coherence between electrical demand and metabolic supply further validates both types of measures as aligned indices of conscious viability.

Thermodynamic modulation across modalities. The impact of temperature on brain function is observed across various measurement domains, from synaptic kinetics to fMRI signal amplitude. Cooling the brain by only a few degrees reduces both spontaneous EEG activity and BOLD signal fluctuations, particularly in regions involved in conscious processing. Electrophysiological recordings show that synaptic latency and refractory periods increase with decreasing temperature, modeled by the above-mentioned Arrhenius-type exponential relationships. These changes result in longer integration windows and a disruption of the millisecond-scale precision needed for gamma-band synchronization. In parallel, metabolic imaging under hypothermic conditions shows a proportional decrease in glucose and oxygen consumption, particularly in gray matter. The relationship between temperature and functional imaging is also evident in the reduction of fMRI global signal variability and resting-state network modularity. This convergence between electrical, metabolic and thermodynamic measurements demonstrates that temperature imposes a hard physical constraint on the neural dynamics required for consciousness. These effects appear consistently across clinical hypothermia studies and intraoperative human data, suggesting their reliability across modalities.

Linking structural and functional connectivity in conscious states. The relationship between structural and functional connectivity provides another layer of cross-modal convergence, especially in the context of conscious processing. Diffusion tensor imaging (DTI) reveals the underlying anatomical pathways through which functional interactions measured by fMRI or EEG must propagate (Yen et al., 2023). Numerous studies demonstrate that consciousness depends not only on the presence of structural connections but also on their dynamic recruitment into coherent activity patterns (Zhang et al., 2017; Zheng et al., 2017). Functional connectivity matrices (FFF) derived from fMRI time series reveal the highest integration within core hubs of the default mode network (DMN) such as the precuneus, medial prefrontal cortex and posterior cingulate regions that are also structurally well-connected. The degree of alignment between structure and function can be assessed using structural-functional coupling indices:

$${\rho }_{SF}={\displaystyle \frac{{\sum }_{i,j}(A_{ij}-\overline{A})(F_{ij}-\overline{F})}{\sqrt{{\sum }_{i,j}(A_{ij}-\overline{A}{)}^2·{\sum }_{i,j}(F_{ij}-\overline{F}{)}^2}}}$$

Where $A\mathrm{a}\mathrm{n}\mathrm{d}F$ are the structural and functional matrices, respectively (Pan et al., 2023; Huang et al., 2023). In conscious states, ${\rho }_{SF}$ tends to be high (~0.5–0.7), whereas in unconscious states (e.g., anesthesia or coma), this coupling decreases markedly (Zarkali et al., 2021; (Rajesh et al., 2024)). Moreover, studies using simultaneous DTI and resting-state fMRI show that reduced structural integrity in long-range white matter tracts correlates with both decreased functional coherence and lowered behavioral responsiveness (Cao et al., 2023). These observations indicate that physical network architecture constrains and partially determines the dynamic patterns of activity associated with consciousness. This means that the anatomy-function relationship operates as a stable cross-modal axis in the evaluation of conscious states.

Integration of peripheral physiology with central measures. A growing body of evidence supports the physiological coupling between peripheral systems—cardiac, respiratory, gastrointestinal—and central neural dynamics relevant to consciousness. Techniques such as electrocardiography (ECG), respiratory plethysmography and electrogastrography (EGG) are now often recorded simultaneously with EEG or fMRI. Heartbeat-evoked potentials (HEPs), measurable by time-locking EEG signals to the R-wave of the ECG, provide a window into cardio-cortical integration. These potentials tend to be larger and more coherent during conscious wakefulness and decrease during deep sleep or under sedation. Cross-modal coherence can be quantified using cross-correlation or mutual information metrics:

$$I(X;Y)=\sum _{x\in X}\sum _{y\in Y}p(x,y)log\left({\displaystyle \frac{p(x,y)}{p\left(x\right)p\left(y\right)}}\right)$$

Where $X\mathrm{a}\mathrm{n}\mathrm{d}Y$ represent peripheral and cortical signal streams, respectively. Higher mutual information values reflect tighter physiological coupling. Similar findings apply to respiratory-entrained oscillations and gut-brain feedback signals, where increased coherence corresponds to more integrated conscious states. These interactions among diverse neurotechnological approaches confirm that conscious states are shaped not only by intracranial variables but also by broader systemic rhythms, offering additional dimensions for their characterization.

Machine learning models of consciousness across modalities. To synthesize data from heterogeneous modalities, recent studies employ machine learning approaches that integrate EEG, fMRI, PET and physiological data into predictive models of consciousness (Garrido Merchán and Molina, 2020; Wang and Ma, 2023; Mohammadi and Ganjtabesh, 2024; Benitez et al., 2024). Features extracted from each modality—such as gamma-band power, functional connectivity matrices, metabolic rates or peripheral variability—are combined into high-dimensional input vectors ${\overrightarrow{x}}_i$. These are used to train classifiers (e.g., support vector machines, random forests, deep neural networks) with conscious state labels (e.g., wake, sleep, anesthesia) as targets. Model accuracy is typically evaluated using k-fold cross-validation (Gu et al., 2017). A linear Support Vector Machine, for instance, minimizes:

$${\min }_{\overrightarrow{w},b}{\displaystyle \frac{1}{2}}\Vert \overrightarrow{w}{\Vert }^2+C\sum _{i=1}^n{\xi }_i\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}{y}_i(\overrightarrow{w}·{\overrightarrow{x}}_i+b)\ge 1-{\xi }_i$$

where $y_i\in \{-\mathrm{1,1}\}$ are consciousness labels and ${\xi }_i$ are slack variables. Models trained on multimodal features consistently outperform those trained on single modalities, achieving accuracy above 90% in discriminating between conscious and unconscious states. Feature importance rankings also tend to reinforce biological findings, with gamma-band coherence, frontoparietal connectivity and CMRglu among the top contributors. These integrative techniques demonstrate that physical parameters drawn from distinct modalities encode shared and robust information about consciousness.

Overall, through formal comparisons and correlation metrics, this chapter illustrated how the integration of different neuro-techniques strengthens the interpretation of consciousness-related data. This consistency validates the use of diverse techniques within a common analytical framework, setting the stage for a multidimensional model of conscious viability.

4. CONSCIOUSNESS WITHIN A MULTIDIMENSIONAL THRESHOLD SPACE

To formalize the concept of consciousness as a state defined by the convergence of multiple biophysical parameters operating within specific threshold ranges, we introduce here a multidimensional state space model in which consciousness emerges from the simultaneous fulfilment of structural, dynamic and energetic constraints. Mathematical formulations are used to define the geometry, topology and temporal stability of this viability space, including sensitivity analyses and critical boundary conditions.

The multidimensional nature of consciousness constraints. The empirical findings presented in previous sections point to the multidimensional character of consciousness. Unlike binary phenomena or linearly scaled variables, consciousness does not emerge from the presence or absence of a single physiological condition. Rather, it reflects the simultaneous alignment of multiple physical constraints, each operating across distinct dimensions: frequency, energy, ion flux, temperature and structural connectivity. These constraints can be conceptualized within a bounded, multidimensional state space $\mathcal{C}\subset {\mathbb{R}}^n$, where each axis corresponds to a normalized, physically grounded parameter $p_i$ and where consciousness is defined as any point $\overrightarrow{p}=(p_1,p_2,...,p_n)\in \mathcal{C}.$ (Tozzi 2019; Kosmyna and Lécuyer, 2019). Mathematically, this region can be represented as an intersection of inequality constraints:

$$\mathcal{C}=\{\overrightarrow{p}\in {\mathbb{R}}^n\mid a_i\le p_i\le b_i,\forall i\in [1,n\left]\right\}$$

Here, $a_i\mathrm{a}\mathrm{n}\mathrm{d}{b}_i$ are empirically derived lower and upper bounds for the i-th parameter. Points falling outside this region correspond to physiologically unsustainable conditions for conscious processes, such as metabolic insufficiency, desynchronized oscillatory activity or ionic collapse. In this model, loss of consciousness results from the transgression of one or more threshold boundaries, reducing the system’s capacity to maintain integrated neural representation. This framework formalizes the idea that consciousness is a property of a physically constrained subspace within the broader physiological state space.

Mathematical formalization of integration and constraint overlap. To extend the model beyond bounded intervals, we consider interdependence between variables. Some parameters influence others directly—for instance, metabolic rate constrains ATP availability, which in turn regulates ion pump efficacy and synaptic firing. These dependencies can be modeled as coupled inequalities or as conditional mappings in a constraint manifold $\mathcal{M}\subset \mathcal{C}$ (Kingston 2020). Define:

$$\mathcal{M}=\{\overrightarrow{p}\in \mathcal{C}\mid g_j(\overrightarrow{p})\le 0,\forall j\in [1,m\left]\right\}$$

where $g_j$ represents a constraint function that encodes nonlinear coupling, such as:

$$g_1(p_{\mathrm{A}\mathrm{T}\mathrm{P}},p_{\mathrm{N}\mathrm{a}},p_{\mathrm{K}})=p_{\mathrm{N}\mathrm{a}\mathrm{K}-\mathrm{A}\mathrm{T}\mathrm{P}\mathrm{a}\mathrm{s}\mathrm{e}}-\gamma ·p_{\mathrm{A}\mathrm{T}\mathrm{P}}·\left({\displaystyle \frac{1}{1+e^{-\lambda (p_{\mathrm{N}\mathrm{a}}-\theta )}}}\right)$$

This reflects that Na⁺/K⁺ pump activity depends on ATP supply and ionic concentration gradients, modulated by sigmoid kinetics. Another example links functional connectivity f to gamma coherence γ and structural integrity sss:

$$g_2(f,\gamma ,s)=f-\alpha ·\gamma ·s$$

These formulations express how localized parameter violations may propagate and disrupt global neural integration. Therefore, the effective conscious state space $\mathcal{M}$ becomes a higher-order constraint surface with system behavior increasingly sensitive near its boundaries. This formalism permits consciousness to be characterized not simply by thresholds, but by their complex interplay and topological relationships.

Parameter sensitivity and critical points. Within the multidimensional constraint surface, some parameters exhibit higher sensitivity, that is, small deviations result in rapid transitions from consciousness to unconsciousness. Sensitivity analysis can be conducted by computing the partial derivatives of a consciousness viability function $\Phi \left(\overrightarrow{p}\right)$, where:

$$\Phi \left(\overrightarrow{p}\right)={\prod }_{i=1}^n{\sigma }_i\left(p_i\right)$$

Each ${\sigma }_i\left(p_i\right)$ is a sigmoidal scaling function mapping parameter values to [0, 1], with 0 representing complete dysfunction. The gradient vector $\nabla \mathsf{\Phi }\left(\overrightarrow{p}\right)$ gives the direction of steepest increase in viability and the magnitude of each partial derivative ${\displaystyle \frac{\partial \mathsf{\Phi }}{\partial p_i}}$ indicates the criticality of the i-th parameter. Parameters such as metabolic rate, gamma coherence and ion gradients show high first-order sensitivity, suggesting that perturbations in these domains exert disproportionate influence on global consciousness viability.

Further, by identifying points ${\overrightarrow{p}}_c\in \mathcal{M}$ such that $\Vert \nabla \mathsf{\Phi }\left({\overrightarrow{p}}_c\right)\Vert \to \infty$, one locates critical transition zones where consciousness becomes unstable. These are analogous to bifurcation points in dynamical systems, representing thresholds beyond which compensatory mechanisms fail (Kaszás et al., 2019). Recognizing these critical points allows researchers to distinguish between parameters that are merely correlated with conscious states and those that are causally foundational.

Dimensional reduction and dominant subspace. Although the conscious state space $\mathcal{M}$ is inherently high-dimensional, evidence suggests that its effective variation may lie in a low-dimensional manifold embedded within the larger space. Principal component analysis (PCA) or singular value decomposition (SVD) can be applied to multivariate physiological data across states to identify orthogonal axes of maximal variance (Brito et al., 2020; Ben Salem and Ben Abdelaziz, 2021). Let $\mathbf{P}\in {\mathbb{R}}^{m\times n}$ be a matrix of m samples with n parameters; PCA yields a decomposition:

$$\mathbf{P}=\mathbf{U}\mathsf{\Sigma }{\mathbf{V}}^T$$

The first few principal components (columns of V) often account for a majority of variance in consciousness-related transitions. These dominant directions define a consciousness manifold ${\mathcal{C}}_r\subset {\mathbb{R}}^r\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}r\ll n$. Empirical studies show that parameters like functional connectivity, cortical metabolism, gamma-band synchrony and K⁺ homeostasis cluster within the first three principal components. The implication is that although consciousness depends on many physiological factors, only a few compound axes explain most of the variability between states.

This dimensional reduction provides a useful simplification for modeling and clinical inference: instead of tracking dozens of biophysical variables, monitoring a small set of key indices may be sufficient to assess the state of consciousness (Perl et al., 2023). The geometry of the subspace also reveals how far a given physiological state lies from the “core” conscious regime. Consciousness, then, becomes not a point, but a stable region within a latent manifold, spanned by physically interpretable vectors.

Temporal stability and dynamical trajectories. Consciousness is not a static condition but a time-evolving process, influenced by endogenous rhythms, environmental stimuli and internal states. To model this temporal behavior, the state vector $\overrightarrow{p}\left(t\right)\in \mathcal{M}$ can be treated as a trajectory through parameter space governed by a dynamical system:

$${\displaystyle \frac{d\overrightarrow{p}}{dt}}=\overrightarrow{F}(\overrightarrow{p},t)$$

Here, $\overrightarrow{F}$ encodes the physiological laws and feedback mechanisms determining how parameters evolve over time. Consciousness corresponds to trajectories that remain inside a stability basin $\mathcal{B}\subset \mathcal{M}$for sufficiently long durations. Transitions into unconsciousness are modeled as escape trajectories crossing the boundary $\partial \mathcal{B}$. These models permit the analysis of resilience, defined by the time-to-boundary under small perturbations $\delta \overrightarrow{p}$. Linear stability analysis using the Jacobian $J={\displaystyle \frac{\partial \overrightarrow{F}}{\partial \overrightarrow{p}}}$ around equilibrium points ${\overrightarrow{p}}_0$identifies whether such points are attractors or saddle nodes (Margazoglou and Magri, 2023; Ahmed et al., 2024):

$$\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}:\mathrm{R}\mathrm{e}\left({\lambda }_i\right(J\left)\right)<0,\forall i$$

Empirical validation of these dynamics comes from anesthesia studies, where consciousness fades over tens of seconds or from seizure models where it collapses in milliseconds. Dynamical modeling thus provides not only spatial, but also temporal constraints on viable conscious regimes, revealing the importance of maintaining trajectories within slow-changing regions of physiological coherence.

Stochastic fluctuations and boundary uncertainty. Since biological systems are inherently noisy, consciousness thresholds are not perfectly sharp. Small stochastic variations in physical parameters can cause the system to hover near the boundary of the conscious manifold $\mathcal{M}$, creating conditions of unstable awareness. To model this, parameters are treated as stochastic processes:

$$d{p}_i={\mu }_i(p_i,t)dt+{\sigma }_i(p_i,t)d{W}_t$$

Here, ${\mu }_i$ is the deterministic drift, ${\sigma }_i$ the diffusion coefficient and $d{W}_t$ a Wiener process. The system evolves as a multidimensional Itô diffusion and the probability of remaining conscious becomes a first-passage problem. The survival function $S\left(t\right)$gives the probability that the system remains within $\mathcal{M}$ up to time t:

$$S\left(t\right)=\mathbb{P}\left(\overrightarrow{p}\right(s)\in \mathcal{M},\forall s\in [0,t\left]\right)$$

These models explain the variability in consciousness seen during transitions (e.g., sedation thresholds or partial awareness in sleep) and suggest that proximity to the manifold boundary increases the likelihood of stochastic exits. The geometry and roughness of $\partial \mathcal{M}$ influence these probabilities, making it essential to account for both deterministic dynamics and stochastic perturbations. These formulations clarify why consciousness may sometimes fluctuate rapidly despite slowly changing physiological inputs.

Overall, by treating consciousness as a constrained trajectory within the space $\mathcal{M}$, this chapter established a unified framework for understanding how diverse physiological processes collectively enable or disrupt conscious experience. This represents a step toward constructing a unified physiological model of consciousness.

5. TOWARD A UNIFIED PHYSICAL FRAMEWORK FOR CONSCIOUSNESS

This chapter integrates the empirical and analytical findings of the previous sections into a unified physical framework for understanding consciousness. We propose that consciousness arises from the intersection of structural, dynamic and energetic constraints, each defined by measurable physiological parameters.

Coherence of structural, dynamic and energetic domains. Converging evidence from electrophysiology, metabolic imaging, connectivity analyses, thermodynamics and peripheral integration supports a view of consciousness as a state governed by the concurrent fulfilment of interdependent physical constraints. At a fundamental level, these conditions fall into three interacting domains: structural (connectivity and architecture), dynamic (oscillations, timing and phase synchrony) and energetic (glucose metabolism, oxygen availability, ATP generation). Each domain provides necessary, though not individually sufficient, support for conscious processes. Structural architecture defines the anatomical substrate for long-range communication, but without dynamic synchronization, this substrate cannot support unified representations. Likewise, energy availability sets the operational limit for synaptic and network activity, but does not prescribe the pattern or content of consciousness. These domains can be represented as overlapping constraint sets $\mathcal{S},\mathcal{D},\mathrm{a}\mathrm{n}\mathrm{d}\mathcal{E}$, with their intersection defining the viable state space for consciousness:

$$\mathcal{C}=\mathcal{S}\cap \mathcal{D}\cap \mathcal{E}$$

Each set $\mathcal{X}\subset {\mathbb{R}}^n$ imposes inequality bounds or dynamic relations over a shared parameter space. The empirical work summarized in previous chapters allows for the parameterization of each domain—such as functional efficiency $E_{glob}>0.3\mathrm{f}\mathrm{o}\mathrm{r}\mathcal{S}$, gamma coherence PLV > 0.4 for $\mathcal{D}$ and CMRglu > 3.5 mg/100g/min for $\mathcal{E}$. The concept of intersection not only identifies the co-dependence of physical variables, but also defines the boundaries of sustainable conscious states. This triangulated model articulates the unified preconditions for consciousness as an emergent product of structurally supported, dynamically synchronized and energetically viable activity.

Embedding temporal dynamics into a spatial framework. Consciousness is not only defined by the fulfillment of spatial constraints at a given time but by the stability of these constraints over relevant temporal windows. A momentary fulfillment of metabolic or synchronization thresholds may be insufficient if not sustained. Thus, the model must incorporate time as a fourth-dimension governing trajectory stability in the multidimensional state space. Formally, let $\overrightarrow{p}\left(t\right)\in {\mathbb{R}}^n$ represent the system’s evolution in parameter space, governed by dynamics ${\displaystyle \frac{d\overrightarrow{p}}{dt}}=\overrightarrow{F}(\overrightarrow{p},t)$. A conscious regime is defined not just as $:\overrightarrow{p}\left(t\right)\in \mathcal{C}$, but as:

$$\forall \tau \in [t,t+\Delta t]:\overrightarrow{p}\left(\tau \right)\in \mathcal{C}$$

for some minimal duration Δt, which may reflect biological constraints on working memory, integration time or neural coding. The requirement of trajectory continuity within $\mathcal{C}$ prevents transient threshold crossings from qualifying as conscious states. Empirical support for this comes from loss-consciousness studies during anesthesia, where brief returns to metabolic sufficiency or oscillatory coherence do not correspond to behavioral responsiveness or subjective reports. Conversely, gradual transitions out of consciousness show early parameter re-entry without cognitive recovery. Consciousness requires continuous occupancy, not merely isolated visits. This approach reconciles the need for both physical sufficiency and temporal persistence in defining viable conscious activity.

Topology of consciousness in parameter space. The geometry of the conscious state space $\mathcal{C}$ reveals further structure when analyzed topologically. Rather than assuming a convex region, empirical fluctuations in physiological parameters suggest that $\mathcal{C}$ may consist of nonlinear, multimodal subregions (Tozzi et al., 2017). For example, slow-wave sleep and light sedation may occupy separate lobes of a partially conscious subspace, with distinct metabolic, oscillatory and structural signatures. To describe this, we treat $\mathcal{C}$ as a union of locally stable basins ${\mathcal{B}}_k$, each with its own centroid ${\mu }_k$ and covariance ${\Sigma }_k$, defined empirically from cluster analyses of high-dimensional data:

$$\mathcal{C}\approx {\bigcup }_{k=1}^K{\mathcal{B}}_k,{\mathcal{B}}_k=\{\overrightarrow{p}\mid (\overrightarrow{p}-{\mu }_k{)}^T{\Sigma }_k^{-1}(\overrightarrow{p}-{\mu }_k)<{\chi }^2\}$$

These ellipsoidal regions describe the local stability of distinct conscious regimes. Transitions between regimes (e.g., light sleep to REM) correspond to inter-basin dynamics, governed by parameter drift and diffusion. Importantly, some unconscious states (deep anesthesia, coma) are associated with attractor basins far removed from any ${\mathcal{B}}_k\subset \mathcal{C}$, making spontaneous recovery unlikely. The topological formulation enables a refined classification of consciousness into clusters of viable configurations, rather than a binary awake/asleep dichotomy. It also suggests that transitions between states are path-dependent, with some trajectories requiring sequential re-entry into multiple constraint basins. This view captures the empirical richness of intermediate states, clarifying the structural nature of conscious continuity and disruption.

Multimodal redundancy and degeneracy. A further dimension of unification involves recognizing that multiple physical mechanisms can fulfill the same functional role in sustaining consciousness. This principle of degeneracy—distinct structures producing the same output—explains why different individuals can maintain awareness under varying physiological configurations (Sajid et al., 2023). For example, both gamma synchrony and beta-gamma coupling can support large-scale network coherence, depending on age, pharmacological state, neural architecture or disease (Liu et al., 2022; Hodnik et al., 2024). Likewise, metabolic resilience may be achieved through increased capillary density in one subject or more efficient mitochondrial function in another. This redundancy is formalized using a many-to-one mapping $f:\mathcal{P}\to \mathcal{F}$, where different parameter vectors ${\overrightarrow{p}}_1,{\overrightarrow{p}}_2,...{\overrightarrow{p}}_n\in \mathcal{C}$ yield the same functional state $f\left({\overrightarrow{p}}_i\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{c}\mathrm{i}\mathrm{o}\mathrm{u}\mathrm{s}.$. Mathematically, the conscious manifold is a fiber bundle over functional equivalence classes:

$$\mathcal{C}={\bigcup }_{y\in \mathcal{F}}{f}^{-1}\left(y\right)$$

This structure allows for individual variability and state plasticity while preserving physical consistency. Redundancy also manifests across measurement modalities; gamma power and BOLD signal, though distinct in mechanism and timescale, often reflect the same underlying parameter class—cortical activation and integration. Recognizing this multimodal mapping prevents false dichotomies between data types and encourages a more abstract, system-level modeling of consciousness as a redundant, emergent product of overlapping physical systems.

Constraint violations and boundary classifications. To fully define the limits of the conscious space, it is essential to characterize what occurs at its boundaries. Each boundary violation represents a specific type of failure, e.g., metabolic collapse, network fragmentation, phase desynchronization, ionic instability, thermal dysregulation. These can be formalized as transitions out of $\mathcal{C}$ via inequality failure: for example, if $p_{\mathrm{g}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}}<{\gamma }_{\mathrm{m}\mathrm{i}\mathrm{n}}$, then $\overrightarrow{p}\notin \mathcal{D}$ and thus $\overrightarrow{p}\notin \mathcal{C}$. Boundary violations can be classified based on their dominant failure mode. Let ${\delta }_i=p_i-b_i$ measure distance to the upper or lower threshold. The primary constraint violation is then:

$$i^{*}=\arg *{\min }_i\left|{\delta }_i\right|\mathrm{s}.\mathrm{t}.p_i\notin [a_i,b_i]$$

This allows the classification of unconscious states not just by behavioral criteria, but by their dominant physical failure. For instance, coma may be categorized by $i^{*}=\mathrm{C}\mathrm{M}\mathrm{R}\mathrm{g}\mathrm{l}\mathrm{u},$, anesthesia by $i^{*}={\mathrm{P}\mathrm{L}\mathrm{V}}_{\gamma }$ and hypothermia by $i^{*}=T$. These physical diagnostics unify different pathologies under a shared formal model and suggest that unconsciousness is not a single state but a collection of parameter-specific collapses.

Overall, this chapter modeled consciousness as a stable region within a multidimensional state space, incorporating both spatial and temporal continuity. It explored the topological organization, redundancy and boundary conditions of this space, providing a coherent system-level account of conscious viability.