Neural Geometrodynamics, Complexity, and Plasticity: A Psychedelics Perspective

Abstract

We explore the intersection of neural dynamics and the effects of psychedelics in light of distinct timescales in a framework integrating concepts from dynamics, complexity, and plasticity. We call this framework neural geometrodynamics for its parallels with general relativity’s description of the interplay of spacetime and matter. The geometry of trajectories within the dynamical landscape of “fast time” dynamics are shaped by the structure of a differential equation and its connectivity parameters, which themselves evolve over “slow time” driven by state-dependent and state-independent plasticity mechanisms. Finally, the adjustment of plasticity processes (metaplasticity) takes place in an “ultraslow” time scale. Psychedelics flatten the neural landscape, leading to heightened entropy and complexity of neural dynamics, as observed in neuroimaging and modeling studies linking increases in complexity with a disruption of functional integration. We highlight the relationship between criticality, the complexity of fast neural dynamics, and synaptic plasticity. Pathological, rigid, or “canalized” neural dynamics result in an ultrastable confined repertoire, allowing slower plastic changes to consolidate them further. However, under the influence of psychedelics, the destabilizing emergence of complex dynamics leads to a more fluid and adaptable neural state in a process that is amplified by the plasticity-enhancing effects of psychedelics. This shift manifests as an acute systemic increase of disorder and a possibly longer-lasting increase in complexity affecting both short-term dynamics and long-term plastic processes. Our framework offers a holistic perspective on the acute effects of these substances and their potential long-term impacts on neural structure and function.

1. Introduction

In this paper, we explore new perspectives to interpret changes in the brain’s landscape and connectivity, focusing on the subtle interplay between structural and dynamical aspects across timescales (fast, slow, and ultraslow). Our primary goal is to present a framework that enhances the understanding of the intricate relationships among brain dynamics, complexity, structure, and plasticity. This framework, which we call “neural geometrodynamics”, draws on principles from non-linear dynamics and is further inspired by conceptual links to general relativity in physics.

In describing neural dynamics, we will refer to the mathematical formalism of neural mass models (NMMs), although other computational neuroscience formulations are equally relevant [1,2]. Neural mass models have been extensively utilized to model various brain activities, from localized brain functions to the coordinated activity observed in different brain regions. By employing mathematical formulations that include essential features like synaptic connectivity and neuronal excitability, NMMs enable the simulation and analysis of complex brain activities in various dynamic regimes [3]. NMMs are particularly useful because they provide a link between the mesoscopic physiological scale and macroscopic brain function, allowing for the connection of effects on neurons at the molecular level, such as those of psychedelics, with those of whole-brain connectivity [4,5].

Analyzing the effects of psychoactive neuroplastogens (psychedelics such as psilocybin or LSD) serves as an illustrative case of the framework, given the immediate and potentially lasting plastic changes these substances can provoke in the brain [6]. By altering neural dynamics and connectivity, psychedelics are thought to induce both transient and sustained shifts in cognition and perception [7]. Several studies underscore the role of psychedelics in inducing neuroplasticity with antidepressant effects, revealing mechanisms at molecular, synaptic, and dendritic levels [8,9], and with significant potential for treating neuropsychiatric disorders [10,11], although the duration and permanence of these effects remain to be fully understood.

Recent conceptual perspectives have enhanced our understanding of the brain’s response to psychedelics, combining biological, dynamical systems, complexity science, and artificial intelligence viewpoints. The REBUS (RElaxed Beliefs Under pSychedelics) framework [12], grounded in the Free Energy Principle (FEP) and the entropic and anarchic brain models, offers a perspective on the effects of psychedelics on the brain whereby psychedelic action results in the collapse of brain functional hierarchies or, in other words, in the “flattening of the landscape” of brain’s dynamics to allow the brain state to escape a deep local minimum. The term annealing is also used in this context in relation to physical annealing in metallurgy and simulated annealing in numerical optimization [13].

Consequently, it has been argued that the observed expansion of the repertoire of functional patterns elicited by hallucinogenic substances can be associated with an increase in entropy in brain dynamics [14,15], with the brain moving to a more disordered state from a relaxation of high-level cognitive priors [12,16]. This may lead to a favorable context for conducting psychotherapy [12,17,18]. Studies on functional neuroimaging regarding psilocybin and LSD effects have shown initial evidence of the mechanistic alterations on brain dynamics at the network level, with the majority of the findings suggesting a relative weakening of usual functional configurations giving place to an increase of brain entropy, global functional integration, and more flexible brain dynamics [14,19,20,21,22,23,24,25,26,27,28]. As mentioned above, these changes are traditionally reflected in the complexity of neural dynamics, which can be evaluated using various techniques such as criticality measures [29,30], complexity measures [31], connectome harmonic decomposition [22,23,24], control theory [25] and Ising (or spinglass) modeling [32,33].

For example, Ising modeling of psychedelics has shown that the increased complexity of brain dynamics under LSD (e.g., increased Ising temperature, Lempel-Ziv, and the Block Decomposition Method complexity) is associated with a decrease of interhemispheric connectivity—especially homotopic links [34], corroborating earlier modeling studies suggesting the central role of long-range connections in controlling phase transitions [35].

The observed push of brain dynamics towards disorder and away from criticality aligns with the REBUS and FEP frameworks, which link the vividness of experience to the entropy of brain activity. At the same time, the notion that a wakeful brain exhibits dimensionality reduction and criticality features that are disrupted by the effect of psychedelics is also predicted by an algorithmic perspective on consciousness [16,36,37], where the psychedelic shift towards disorder is associated with a disruption of the world-modeling/world-tracking circuits in the brain.

Another feature of brain dynamics related to the collapse of higher-order cognitive functions under psychedelics in the REBUS framework is the hierarchical organization along the uni- to trans-modal functional gradient [38]. This asymmetry in neural activity reflects the bottom-up and top-down information flows in cognitive processing [39,40]. This has been suggested to be intimately linked to non-equilibrium dynamics in thermodynamic-inspired frameworks where the level of hierarchy is related to the amount of brain signal irreversibility as well as entropy production [41,42,43]. Indeed it has been demonstrated that the principal functional gradient collapses under the influence of various psychedelics [44,45,46].

A related perspective for this paper is the CANAL framework [11] for describing the pathological plasticity of “being stuck in a rut” in certain mood disorders and the potential therapeutic role of psychedelics through the concept of metaplasticity. In contrast to psychedelics, these changes are reflected in neural dynamics with brain signatures of excessively rigid and highly ordered functional states [47]. The CANAL framework has been further extended by establishing connections with deep artificial neural networks (Deep CANAL [48]) to introduce a distinction between two distinct pathological phenomena—one related to fast brain dynamics and their slow and ultraslow counterparts. These distinctions will be naturally integrated into the presented framework (see the Appendix A Figure A1 relating the concepts in the different frameworks).

While the discussion is centered on the effects of psychedelics, the framework proposed here extends more generally to other phenomena related to plasticity, including neurodevelopment, pathological plasticity in mood disorders [49], and interventions that alter brain dynamics like transcranial brain stimulation (tES) [50], transcranial magnetic stimulation (TMS), or electroconvulsive therapy (ECT).

In what follows, we formalize the notions of brain dynamics, plasticity, and their associated timescales and subsequently use them to study the impact of psychedelics on the brain. In the last section, we draw connections between the framework and concepts from general relativity in physics. We hope these parallels will illuminate the complex relationship between the structure and function of brain dynamics. Figure 1 illustrates the reciprocal dynamics between brain states and connectivity as conceptualized in the neural geometrodynamics framework.

2. Dynamics across Timescales

The state of a system can be defined by a set of coordinates in phase space: a multidimensional manifold in which each dimension corresponds to one of the variables. For a single particle moving in one dimension, the phase space is two-dimensional, with one axis representing its position and the other representing its momentum. For example, Figure 2 illustrates the phase space of a pendulum with friction. In phase space, and perhaps after some transient period, the possible trajectories of the states of the system lie in a reduced or invariant manifold (an attractor, see Box 1 for a glossary of terms), which we may refer to as the “geometry” or latent “structure” of the phase space. Together, the structure (geometry and topology) of the phase space with its invariant properties can be referred to as the dynamical landscape, where the depth or shallowness of the “valleys” can, in some cases, be interpreted as the stability of the state in that location given some stochastic forcing. For example, in mechanics, the landscape can be labeled by potential energy isolines, e.g., in a physical system such as in the pendulum example in Figure 2 (bottom right), or their generalization, Lyapunov functions [51]. For definitions and explanations of key terms used in this discussion, see the Glossary in Box 1.

Box 1. Glossary.

State of the system: Depending on the context, the state of the system is defined by the coordinates x (Equation (1), fast time view) or by the full set of dynamical variables (x, w, $\theta$)—see Equations (1)–(3).

Entropy: Statistical mechanics: the number of microscopic states corresponding to a given macroscopic state (after coarse-graining), i.e., the information required to specify a specific microstate in the macrostate. Information theory: a property of a probability distribution function quantifying the uncertainty or unpredictability of a system.

Complexity: A multifaceted term associated with systems that exhibit rich, varied behavior and entropy. In algorithmic complexity, this is defined as the length of the shortest program capable of generating a dataset (Kolmogorov complexity). Characteristics of complex systems include nonlinearity, emergence, self-organization, and adaptability.

Critical point: Dynamics: parameter space point where a qualitative change in behavior occurs (bifurcation point, e.g., stability of equilibria, emergence of oscillations, or shift from order to chaos). Statistical mechanics: phase transition where the system exhibits changes in macroscopic properties at certain critical parameters (e.g., temperature), exhibiting scale-invariant behavior and critical phenomena like diverging correlation lengths and susceptibilities. These notions may interconnect, with bifurcation points in large systems leading to phase transitions.

Temperature: In the context of Ising or spinglass models, it represents a parameter controlling the degree of randomness or disorder in the system. It is analogous to thermodynamic temperature and influences the probability of spin configurations. Higher temperatures typically correspond to increased disorder and higher entropy states, facilitating transitions between different spin states.

Effective connectivity (or connectivity for short): In our high-level formulation, this is symbolized by w. It represents the connectivity relevant to state dynamics. It is affected by multiple elements, including the structural connectome, the number of synapses per fiber in the connectome, and the synaptic state (which may be affected by neuromodulatory signals or drugs).

Plasticity: The ability of the system to change its effective connectivity (w), which may vary over time.

Metaplasticity: The ability of the system to change its plasticity over time (dynamics of plasticity).

State or Activity-dependent plasticity: Mechanism for changing the connectivity (w) as a function of the state (fast) dynamics and other parameters ($\alpha$). See Equation (2).

State or Activity-independent plasticity: Mechanism for changing the connectivity (w) independently of state dynamics, as a function of some parameters ($\gamma$). See Equation (2).

Connectodynamics: Equations governing the dynamics of w in slow or ultraslow time.

Fast time: Timescale associated to state dynamics pertaining to x.

Slow time: Timescale associated to connectivity dynamics pertaining to w.

Ultraslow time: Timescale associated to plasticity dynamics pertaining to $\theta =(\alpha ,\gamma )$—v. Equation (3).

Phase space: Mathematical space, also called state space, where each point represents a possible state of a system, characterized by its coordinates or variables.

Geometry and topology of reduced phase space: State trajectories lie in a submanifold of phase space (the reduced or invariant manifold). We call the geometry of this submanifold and its topology the “structure of phase space” or “geometry of dynamical landscape”.

Topology: The study of properties of spaces that remain unchanged under continuous deformation, like stretching or bending, without tearing or gluing. It’s about the ‘shape’ of space in a very broad sense. In contrast, geometry deals with the precise properties of shapes and spaces, like distances, angles, and sizes. While geometry measures and compares exact dimensions, topology is concerned with the fundamental aspects of connectivity and continuity.

Invariant manifold: A submanifold within (embedded into) the phase space that remains preserved or invariant under the dynamics of a system. That is, points within it can move but are constrained to the manifold. Includes stable, unstable, and other invariant manifolds.

Stable manifold or attractor: A type of invariant manifold defined as a subset of the phase space to which trajectories of a dynamical system converge or tend to approach over time.

Unstable Manifold or Repellor: A type of invariant manifold defined as a subset of the phase space from which trajectories diverge over time.

Latent space: A compressed, reduced-dimensional data representation (see Box 2).

Topological tipping point: A sharp transition in the topology of attractors due to changes in system inputs or parameters.

Betti numbers: In algebraic topology, Betti numbers are integral invariants that describe the topological features of a space. In simple terms, the n-th Betti number refers to the number of n-dimensional “holes” in a topological space.

Box 2. The manifold hypothesis and latent spaces.

The dimension of the phase (or state) space is determined by the number of independent variables required to specify the complete state of the system and the future evolution of the system. The Manifold hypothesis posits that high-dimensional data, such as neuroimaging data, can be compressed into a reduced number of parameters due to the presence of a low-dimensional invariant manifold within the high-dimensional phase space [52,53]. Invariant manifolds can take various forms, such as stable manifolds or attractors and unstable manifolds. In attractors, small perturbations or deviations from the manifold are typically damped out, and trajectories converge towards it. They can be thought of as lower-dimensional submanifolds within the phase space that capture the system’s long-term behavior or steady state. Such attractors are sometimes loosely referred to as the “latent space” of the dynamical system, although the term is also used in other related ways. In the related context of deep learning with variational autoencoders, latent space is the compressive projection or embedding of the original high-dimensional data or some data derivatives (e.g., functional connectivity [54,55]) into a lower-dimensional space. This mapping, which exploits the underlying invariant manifold structure, can help reveal patterns, similarities, or relationships that may be obscured or difficult to discern in the original high-dimensional space. If the latent space is designed to capture the full dynamics of the data (i.e., is constructed directly from time series) across different states and topological tipping points, it can be interpreted as a representation of the invariant manifolds underlying system.

2.1. Fast Time: Neural Dynamics

Here, we discuss the first equation in neural geometrodynamics in the context of neural mass models, though the ideas are applicable more extensively in computational neuroscience. The standard equation we use in neural mass modeling is a multidimensional ODE of the form

$$\dot{x}=f\left(x;w,\eta \left(t\right)\right)$$

(1)

with $x\in {\mathbb{R}}^n$ and where w denotes connectivity parameters and where, as usual, a dot over a variable denotes its time derivative. The above equation governs dynamics at short time scales (seconds or less) when the connectivity parameters w are assumed to be constant.

The external input term $\eta \left(t\right)$ makes the equations non-autonomous (an autonomous ODE does not explicitly depend on time). This term can refer to external forces providing random kicks to the trajectory or to a more steady and purposeful forcing from unspecified internal systems, external inputs from sensory systems, or external electric fields, for example.

We may think of this equation describing phenomena in fast time scales as providing the “structure” for the dynamics of the neuronal population state. The fast timescale is set by synaptic transmission (milliseconds) and by ephaptic coupling (electromagnetic waves) [56,57,58,59] on a nanosecond or subnanosecond scale [58].

Equation (1) characterizes the dynamical landscape, which is established through the geometric structure of the phase space, where trajectories are shaped by the given set of ordinary differential equations. The landscape is determined by the functional form of $f(x;w,\eta (t\left)\right)$ and by the parameters w, and is analogous to the Neural Activation Landscape proposed in [48]. In the REBUS model [12], w would correspond to the weights or precision assigned to priors/beliefs from the Free Energy perspective, while from the Entropic Brain perspective w would correspond to the weights of the effective connectivity between neuronal populations on the macroscopic scale.

More specifically, we talk about the landscape as defined by the manifold generated by the motion of trajectories with coordinates $x\in {\mathbb{R}}^n$. Typically, trajectories lie in a reduced manifold of dimensionality lower than ${\mathbb{R}}^n$. The fact that such a reduced space exists means that it can be generated by coordinates in a reduced latent space. The geometry and topology of this reduced space in different states provide a synthetic description of the dynamics and are of special interest [37].

2.2. Slow Time: Connectodynamics

The landscape, like that on planet Earth, may appear to be static; however, in reality it is not fixed. It flows in slow time; thus, we can consider changes in connectivity in the system, which is now $w=w\left(t\right)$. We refer to the potential for such changes as the plasticity of the system. The general form of this equation is $\dot{w}=g(x,w;\theta )$, with $\theta$ standing for the set of parameters controlling the plasticity.

To be more concrete, we can think of two types of process: one that modifies the connectivity parameters independently of the system’s state ($\psi$), and another that is a function of the state (e.g., Hebbian plasticity [60], h). We express this by writing

$$\dot{w}=\psi (w;\gamma )+h(x,w;\alpha ).$$

(2)

(with the second term understood as not separable into parts where any part is a function of only w). This decomposition separates out state-dependent (via the term $h(x,w;\alpha )$) and state-independent plasticity (with $\psi (w;\gamma )$) processes. The set of parameters $\theta$ is similarly decomposed as $\theta =(\alpha ,\gamma )$: we separate out the plasticity-controlling parameters in order to differentiate the state-dependent (α) and state-independent (γ) plasticity control parameters (e.g., Hebbian vs. drug-enhanced structural plasticity [11]). The parameters $(\alpha ,\gamma )$ may vary in time to reflect, for example, the effects of drugs. The dynamics of these parameters are formalized in the next section.

Hebbian plasticity is the most prominent example of state-dependent plasticity [60]. State dependence implies that state-related concepts such as system temperature, phase transitions, and critical phenomena are relevant for the study of the dynamics of plasticity. In particular, within the scope of slower “slow time” (taking place over many hours), we include homeostatic plasticity [61,62,63,64], which may itself target desired complexity states as a homeostatic goal [65,66]. In the case of state-independent plasticity, there are numerous candidates for these plastic processes, such as heterosynaptic plasticity [67] or critical-period plasticity [68].

In summary, the functions h and $\psi$ with parameters $\alpha$ and $\gamma$ regulate connectodynamics, defining where and how fast the effective connectivity will change in a state-dependent or state-independent way.

These connectodynamics differential equations define a new dynamical landscape, which we can call the plasticity landscape (analogous to the Synaptic Weight Landscape in [48]). The state w in this plasticity landscape will determine the shape of the neural dynamics landscape.

2.3. Ultraslow Time: Metaplasticity

Plasticity is required to adapt to a changing environment [69], and the environment may change at different rates at different times. Plasticity in the healthy brain should match this variation in the character of dynamics accordingly. This is analogous to the situation in biology, where optimal mutation rates ensure successful adaptation in a tradeoff with genetic integrity [70]. More specifically, the plasticity-regulating parameters $\alpha$ and $\gamma$ in Equation (2) should adapt to changes in the environmental conditions.

In pathological cases, plasticity levels can either become overly exuberant, reflecting the notion of catastrophic forgetting in artificial neural networks, or impoverished and rigid, reflecting general plasticity loss [48]. These scenarios can be tentatively related to certain neurological and psychiatric conditions. For example, reduced plasticity could underlie conditions such as major depressive disorder, obsessive-compulsive disorder, anxiety, or substance abuse [48,71].

To account for the dynamics of plasticity, we allow the plasticity parameters to be dynamic, i.e.,

$$\begin{array}{c}\hfill \dot{\theta }=\xi (x,w,\theta ;\mu \left(t\right))\end{array}$$

(3)

This equation is again state-dependent, allowing the system to respond to changes in the neural dynamics (with state dynamics as drivers of plasticity parameter regulation [72]), including critical phenomena (changes in criticality regime [65]) and complexity. Plasticity dynamics reflect changes in the parameters regulating state-dependent (Hebbian) plasticity (changes in $\alpha$) during neurodevelopment, and state-independent plasticity, such as the ones induced by psychedelics in the acute or post-acute phases (changes in $\gamma$). Finally, this equation is a function of other parameters and non-autonomous terms ($\mu \left(t\right)$), reflecting external perturbations of the system, such as those from drugs. We provide analogies in the context of sailing and electrodynamics in Appendix A to further clarify these concepts.

The dynamics of plasticity presented above reflect a physiological principle well described by Abraham et al. in the definition of metaplasticity [73]:

Metaplasticity […] is manifested as a change in the ability to induce subsequent synaptic plasticity, such as long-term potentiation or depression. Thus, metaplasticity is a higher-order form of synaptic plasticity.

[73]

Thus, metaplasticity and its counterparts are terms used in neuroscience to refer to the plasticity of synaptic plasticity. That is, the idea that the ability of synapses to strengthen or weaken in response to increases or decreases in their activity (which is called synaptic plasticity) can be modulated based on the history of the synaptic activity or other factors (e.g., age, neuromodulatory systems, drugs, or lifestyle [74]). Metaplasticity has important implications for the learning and memory of an organism, as it can regulate the ability of synaptic plasticity to change and adapt over time as required by its environmental context.

We call the set of Equations (1)–(3)—somewhat whimsically—the equations for neural geometrodynamics in reference to the equations of general relativity in physics. We recall that general relativity provides equations defining the dynamics of spacetime geometry (via the “metric”) coupled with matter [75]. Section 4 elaborates further on this parallel.

3. Dynamics under Psychedelics

Psychedelics like psilocybin and LSD act as agonists or partial agonists for serotonin 5-hydroxytryptamine 2A (5-HT_2A) receptors, specifically targeting Layer V cortical pyramidal neurons [11,12,14,76,77], leading to increased neuronal excitability through an increase in excitatory postsynaptic currents and discharge rates in pyramidal neurons [12]. The highest expression of 5-HT2ARs is found on the apical dendrites of Layer 5 pyramidal cells in both cortical and subcortical structures [12,78]. In the cortex, 5-HT_2A receptors are strongly expressed along a steep anteroposterior gradient [79]. When psychedelics bind to these receptors, they can lead to a gradual increase of the excitability of these pyramidal neurons—depolarizing them and making them more susceptible to excitatory inputs such as those associated with glutamate receptors [79]—much as the gain knob in an amplifier. This increased excitability and susceptibility to inputs can lead to changes in the firing patterns of these neurons and alterations in the overall neural circuit activity. Recognized for their potent and immediate impact on the brain, these drugs cause a swift reconfiguration of neural dynamics. As we explain, this immediate effect is represented in our model by state-independent alterations in the connectivity parameters (w) (see Figure 3).

How are these effects represented in Equations (1) and (2)? If we include the neuromodulatory nodes in our model—the dorsal raphe and median raphe nuclei in the brainstem are the source of most serotonergic neurons projecting throughout the brain [79]—, the modulation of serotonin receptors could be represented by changes in neuromodulatory connectivity (the subset of w parameters in the model connecting the raphe nuclei to other nodes). Alternatively, if neuromodulatory nodes are not explicitly included in the model, for the purposes at hand, we can think of the changes in the excitability of the nodes affected by neuromodulatory inputs as leading to changes in their effective connectivity (w) to other nodes (e.g., through an increase of the connectivity of glutamatergic synapses into Layer 5 pyramidal cells).

The abrupt shift induced by psychedelics can be thought of as a transformation of the phase space geometry that allows the neural state to explore new trajectories. This process manifests in an increase of complexity and disorder, which can be measured using various tools in different modalities, e.g., EEG or fMRI BOLD, and with measures such as entropy, fractal dimension, algorithmic complexity, etc. [29,31,34,80]). The decrease in effective connectivity under LSD (especially in interhemispheric homotopic connections), as inferred using Ising modeling of BOLD signals measured using fMRI imaging, is associated with a subsequent increase in algorithmic complexity [34].

Psychedelic-induced changes in connectivity can correspond to a flattening of the dynamical landscape [12] or a destabilization of it [48].In our framework, the alteration of effective connectivity results in an immediate and state-independent remodeling of the dynamical landscape during the acute phase of psychedelics, which is represented by the term $\psi (w;\gamma )$ in the connectodynamics equation (Equation (2)). Notably, in the REBUS model and the Entropic Brain perspective [12], the weights of the effective connectivity during the psychedelic-induced state are “flattened” or “de-weighted”, representing a more symmetrical and non-hierarchical connectivity profile.

The instantaneous modification of the landscape is, however, ephemeral, gradually fading as the acute effects of the psychedelic wears off. The system returns to near its original geometrical configuration, though with lasting influences brought about by the plastic changes resulting from the exploration of new trajectories in the acute phase. These residual changes are captured by the state-dependent plasticity term $h(x,w;\alpha )$, which reflects changes in connectivity due to Hebbian plasticity that arise from the co-activation of neurons during the acute psychedelic stage.

In the literature, there is an increasing body of evidence suggesting a post-acute phase following psychedelic exposure characterized by a period of enhanced plasticity [7,11,12,81,82]. This phase can be interpreted as an extended window of malleability of the landscape, which could have profound implications for learning and therapy. Such a window of plasticity has been related to increased neurogenesis and upregulation of Brain-Derived Neurotropic Factor (BDNF) in humans and mice [8]. The activity-dependent release of BDNF plays a crucial role in selectively strengthening active synapses while weakening inactive ones, a critical process for Hebbian-type plasticity. Intriguingly, recent studies with mice have found psychedelic-induced changes in plasticity and antidepressant-like behavior dependent on the increase of endogenous BDNF and TrkB binding (the receptor of BDNF) that are independent of the activation of 5-HT_2A [9,10].

In terms of our model, these two pathways correspond to changes in connectivity through Equation (2) due to a temporary modulation of the parameters $\gamma$ and $\alpha$ (i.e., metaplasticity; see Equation (3)) that upregulates state-independent and state-dependent plasticity processes, respectively. The strong acute-phase increase of state-independent plasticity ($\psi$) would be directly associated with the activation of serotonergic receptors, as discussed above, with a possible gradual decrease during the post-acute phase (the solid white line in Figure 3). The sustained increase of state-dependent plasticity (h) in the post-acute phase (the dashed black line in Figure 3) would be linked to dendritic growth, neurogenesis, upregulation of BDNF, and other related changes. This means that in the post-acute period the landscape would be more responsive to state changes (itself being influenced by external factors), offering a potential mechanism for the long-lasting changes reported after psychedelic experiences. Such external influences are modeled by the external input term $\eta \left(t\right)$ in the state equation (Equation (1)), and can represent environmental/sensory inputs, psychotherapy, or neuromodulatory brain stimulation techniques such as transcranial electrical current stimulation (tES).

Dynamics of Psychedelics and Psychopathology

Recently, psychedelic medicine has emerged as a promising direction for treating mental disorders such as depression and addiction [83]. The nuanced interaction between the brain’s neurophysiology and emergent brain activity underlies the pathophysiology of mood disorders, often resulting in a persistent and maladaptive rigidity in cognitive and emotional processes [84]. Such changes to the brain’s neurophysiology can be explained through the CANAL framework, whereby pathological plasticity, often caused by a traumatic event asserts itself and dominates brain activity, driving the brain state to become “stuck in a rut” [11], i.e., a deepening minimum in the dynamical landscape (see Figure 4).

The interplay between external inputs, neural (fast time) dynamics, and connectivity (slow time) dynamics can drive the system into a joint stable and canalized state of lower complexity. Under the influence of psychedelics, more diverse and complex dynamics destabilize the plasticity equilibrium point, leading to a more fluid and adaptable neural state in a process that is amplified by the plasticity-enhancing effects of psychedelics. This shift manifests as an acute systemic increase of disorder, and possibly as a longer-lasting increase in complexity (Ising temperature, Lempel-Ziv complexity, etc.) that affects both short-term dynamics and long-term plastic processes.

The CANAL framework offers insight into the neural mechanisms underlying the persistence of various brain disorders. In particular, psychedelics may mediate their effects by altering the balance between stability and plasticity in neural networks through metaplasticity, and thereby act as potential therapeutic treatments. By acting on the serotonergic receptors, they trigger a cascade of neurochemical events that subsequently facilitate the reorganization of entrenched neural patterns. As discussed above, this alteration of the neural network during the acute phase (connectodynamics) can be interpreted as a rapid deformation or flattening of the landscape that allows the trapped state to escape and access more adaptive cognitive and emotional patterns. The rapid increase in complexity (a change in the dynamics) is itself a likely driver of metaplasticity. The acute phase is believed to be followed by an extended window of malleability of the landscape, otherwise known as a “window of plasticity”, where treatments such as psychotherapy and transcranial electrical stimulation can further alter the pathological rigidity characteristic of various brain disorders (see Figure 4).

4. Neural Geometrodynamics and General Relativity

A parallel can be drawn between neural geometrodynamics and Einstein’s equations of general relativity—the original geometrodynamics. Both frameworks involve the dynamical interaction between structure and resulting activity, each influencing and being influenced by the other. The Einstein field equations, including the cosmological constant $\Lambda$, are

$$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R+\Lambda g_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }.$$

(4)

Here, $g_{\mu \nu }$ is the metric tensor, $R_{\mu \nu }=R_{\mu \nu }\left[g_{\mu \nu }\right]$ is the Ricci curvature tensor and a function of $g_{\mu \nu }$, $R\left[g_{\mu \nu }\right]$ is the Ricci scalar (or curvature scalar) and a function of $g_{\mu \nu }$, and $T_{\mu \nu }$ is the stress–energy tensor, a function of the mass and energy distribution (all the indices refer to spacetime dimensions), which is sometimes called the energy–momentum tensor. This is a central concept in general relativity that encapsulates the distribution and flow of energy and momentum in spacetime; its components include the energy density, momentum density, and stress (pressure and shear stress) within a given region. In addition, G is the gravitational constant, c is the speed of light, and $\Lambda$ is the cosmological constant. These equations describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. Specifically, they equate local spacetime curvature (on the left-hand side) with the local energy and momentum within that spacetime (on the right-hand side).

To complete these equations, the geodesic equation portrays how particles (matter) move in this curved spacetime, encapsulated by the notion that particles follow the straightest possible paths (geodesics) in curved spacetime:

$$\frac{d^2{x}^{\mu }}{d{\tau }^2}+{\Gamma }_{\rho \sigma }^{\mu }\frac{d{x}^{\rho }}{d\tau }\frac{d{x}^{\sigma }}{d\tau }=0$$

(5)

where $x^{\mu }$ are the coordinates of the particle, $\tau$ is the proper time along the particle’s path, and ${\Gamma }_{\rho \sigma }^{\mu }\left[g_{\mu \nu }\right]$ are the Christoffel symbols, which are a function of $g_{\mu \nu }$ and encode the connection (a mathematical object that describes how vectors change as they are parallel transported along curves in spacetime). The stress–energy tensor $T_{\mu \nu }$ can be computed from the state of the particles to close the system of equations. For example, for N particles it is provided by $T^{\mu \nu }={\sum }_i{m}_i{u}_i^{\mu }{u}_i^{\nu }\delta (\mathbf{x}-{\mathbf{x}}_i)$, where $m_i$ and $u_i$ are the mass and velocity of the ith particle. More generally, the stress–energy tensor represents the state of matter and energy, which corresponds to x in our neural model. The metric $g_{\mu \nu }$, which specifies the geometry of spacetime, is akin to the connectivity w, which shapes the structure of the space where fast dynamics occur.

In the context of neural mass models, the state equation $\dot{x}=f(x;w)$ is analogous to the geodesic equation: “the state of the system evolves according to the landscape geometry specified by the parameters w”. On the other hand, the connectodynamics equation $\dot{w}=h(x,w;\theta )$ (with $\theta$ standing for plasticity parameters) is analogous to Einstein’s field equations; the parameters w, which describe the “structure” of the space where dynamics take place, evolve according to the current state of the system x and its ‘readiness’ for plasticity (parametrized by $\theta$).

The analogy to psychedelic effects in general relativity can be clarified further. The neural effects of psychedelics, as we understand them, start with a disruption of connectivity in a spatially dependent manner. Since the analog of w is g (the metric), in cosmological terms, we would first see a dynamic deformation of spacetime independent of the mass distribution (state-independent plasticity). Spacetime would “flatten”. This would cause the mass in the universe to escape from gravitational wells following new geodesics (just as the state in the brain will explore new regions of phase space), in turn creating further deformations of spacetime (state-dependent plasticity).

We emphasize that this comparison is largely metaphorical and therefore limited: the mutual influence between particles and spacetime in general relativity is akin to the state of the neural system and its underlying connectivity parameters. In both cases, dynamics and structure are intertwined (see Figure 5). However, as an example of the limitations of the analogy, the slow and fast nature of the different variables is interchanged in the two formulations, with spacetime responding faster (at the speed of light) to changes in the distribution of energy than the stress-energy tensor itself.

4.1. Metaplasticity and Variable Constants in Cosmology

In our neural mass model framework, the concept of metaplasticity is introduced to refer to dynamic variations in the plasticity control constants, in the connectodynamics equation, namely, $\theta$. This set of constants can be represented as evolving over time as a function of the state of the system or other relevant variables:

$$\begin{array}{c}\hfill \dot{\theta }=\xi (x,w,\theta ;\lambda ),\end{array}$$

(6)

where $\xi$ defines the evolution of the plasticity control constants with parameters $\lambda$.

Analogously, in the realm of general relativity and cosmology, it has been speculated that the fundamental constants such as the speed of light c, gravitational constant G, or cosmological constant $\Lambda$, may in fact be dynamic. Although not part of the mainstream cosmological model, theories proposing variable constants such as the “Variable Speed of Light” (VSL) or “Variable Cosmological Constant” provide an intriguing parallel. For instance, within VSL theories the speed of light c is postulated to vary over cosmological time scales. Certain hypothetical dynamical equations could dictate the dynamical evolution of these constants. Although these theories are quite speculative and do not form a part of mainstream physics, they offer an interesting perspective on the concept of metaplasticity and its potential implications for the dynamical evolution of neural mass models and the structure of their landscapes.

4.2. Psychedelics as Wormholes in the Neural Landscape

In the parallel between general relativity and neural geometrodynamics, we see the effects of psychedelics as a deformation of the neural landscape (spacetime) that allows the brain state (of a particle or set of particles) to escape from a local minimum and transition to another location in the landscape (spacetime). Although transitions may be smooth and respect the topology of the landscape [85,86], deformations of the landscape may be more extreme as well, i.e., sharp transitions through a topological tipping point of the dynamical landscape. The creation of a wormhole (in 4D or higher dimensional spaces) would alter the topological structure of the manifold it inhabits and the associated Betti numbers. This may be due to external inputs ($\eta \left(t\right)$) when the system is non-autonomous [87], e.g., due to sensory or brain stimulatory effects. As we have discussed, this may be due to connectivity dynamics.

Wormholes, a term coined by John A. Wheeler [88] (sometimes called Einstein–Rosen bridges), are solutions to the Einstein field equations of general relativity which certain models suggest could exist under specific conditions. The creation of a wormhole in general relativity can be viewed as a profound deformation of spacetime, bending and connecting distant parts of the universe in such a way that matter/energy (such as an astronaut) can travel vast distances in an instant. This change in the geometry and topology of spacetime can be likened to the effect of psychedelics on the human mind. Just as a wormhole alters the structure of spacetime, psychedelics may radically deform the dynamical landscape of neural dynamics, creating connections across distant landscape locations. In the same way that the astronaut would use a wormhole to bypass vast stretches of space, the deformation caused by psychedelics may allow the state of the brain to “tunnel out” and escape from a local minimum or stuck pattern of thought, providing access to new areas of the landscape, new perspectives, and potentially unexplored territories of consciousness. Both phenomena are characterized by a fundamental transformation that enables traversal into otherwise inaccessible regions, whether in physical space or the brain’s dynamical landscape; see Figure 6 for a sketch of this concept.

4.3. Characterizing the Landscape

An important challenge in the program of neural geometrodynamics is to explore practical methods with which to characterize the landscape. Here, we can again draw inspiration from physics and mathematics.

The roots of this approach can be traced back to the 19th century, when Carl Friedrich Gauss pioneered the field of differential geometry. Gauss’s Theorema Egregium demonstrated that the curvature of a surface could be determined entirely by measurements within the surface, without any reference to the surrounding space [89]. This seminal insight has laid the groundwork for understanding manifolds in various contexts, including the theory of relativity. In the era of general relativity, the interplay between geometry and physics was further enriched. Differential geometry and algebraic topology, which comes into play when one is interested in the global properties of the manifold, such as its shape, connectedness, and the presence of holes [90,91], became essential in describing the fabric of spacetime itself. It enabled physicists to conceptualize how mass and energy warp the geometry of spacetime, thereby influencing the motion of objects.

In our current endeavor, these ideas find application in characterizing the complex dynamical landscapes of neural data. Modern tools from deep learning, such as variational autoencoders, can be used to unravel the reduced spaces underlying neuroimaging or neurophysiological data [54,55], while dynamical systems theory in concert with differential geometry, group theory, and algebraic topology data analysis [92] offer robust frameworks to understand and characterize them [87,93,94,95,96,97]. Topological data analysis can also be used to explore the graphs associated with model space, for example, the structural (connectome) or effective connectivity between regions in the brain (see [98] for a recent review). Topological methods have already been successfully employed to analyze detailed microscopic models [95], to study the relationship of criticality and topology in models [99], and to characterize functional brain networks derived from neuroimaging data [85,86,98].

World-tracking constraints force the brain as a dynamical system to mirror the symmetry in the data [37], a requirement that translates into constraints on structural and dynamical aspects of the system (and which can be analyzed using Lie group theory). This suggests leveraging the known links between topology and Lie groups [100]. The convergence of these mathematical techniques extends to neuroscience the fruitful exercise in physics of linking geometry and topology.

Finally, it would be interesting to explore if hierarchical data processing systems such as the brain display dynamical manifolds with hierarchical structure, including topology. This possibility is intuitive given the connections between the notions of criticality, information processing, and hierarchical organization [34,101]. In this sense, the effects of psychedelics, which are seen to increase the temperature of the system [34] and the complexity of dynamics, should be reflected as an increase in the topological complexity of the associated dynamical attractors, as we discussed above with the analogy to wormholes.

The relationship between hierarchy and topological complexity could be analyzed, for example, by exploring artificial neural networks carrying out hierarchical processing (any generative deep network trained on real-world data would do, in principle). Such networks could then be used to generate neural activation data and analyze, for instance, whether the depth of the network (the number of layers in its hierarchical architecture) is reflected in the topology (e.g., in Betti numbers) associated with the data or its latent space.

5. Conclusions

In this paper, we have defined the umbrella of neural geometrodynamics to study the coupling of state dynamics, their complexity, geometry, and topology with plastic phenomena. We have enriched the discussion by framing it in the context of the acute and longer-lasting effects of psychedelics.

As a source of inspiration, we have established a parallel with other mathematical theories of nature, specifically, general relativity, where dynamics and the “kinematic theater” are intertwined.

Although we can think of the “geometry” in neural geometrodynamics as referring to the structure imposed by connectivity on the state dynamics (paralleling the role of the metric in general relativity), it is more appropriate to think of it as the geometry of the reduced phase space (or invariant manifold) where state trajectories ultimately lie, which is where the term reaches its fuller meaning. Because the fluid geometry and topology of the invariant manifolds underlying apparently complex neural dynamics may be strongly related to brain function and first-person (structured) experience [16], further research should focus on creating and characterizing these fascinating mathematical structures.