№ 5 — Analogue Cognition · BÄRŌ DYNAMICS

§ 01 · Purpose

A living scientific map, not an explainer

№ 5 is built to make a novel theory of consciousness — Robert Worden's Projective Wave Theory of Consciousness (PWTC) and the Requirement Equation — inspectable: every claim is rendered as a named equation pinned to the anatomical structure that implements it, sitting beside the established neuroscience scaffold the theory has to interoperate with (predictive coding, free energy, Hopfield CA3, HPA feedback, somatic markers, global-workspace ignition, metacognition, allostasis, communication policy, travelling waves, sleep-replay consolidation, drift-diffusion).

The goal is not pedagogy. The goal is to give scientists a single page where they can read the math, click the brain, watch the layers integrate, and decide whether the model's predictions are worth testing against real neural recordings.

§ 02 · Anatomical substrate

What the model is rendered on

The cortex you see is not stylised. It is the standard HCP cortical parcellation drawn in 3-D from the same coordinate system used in the connectomics literature.

Cortex

HCP-MMP1.0 multi-modal parcellation, 360 parcels (180 per hemisphere). Glasser, M. F., Coalson, T. S., Robinson, E. C., et al. (2016). A multi-modal parcellation of human cerebral cortex. Nature, 536, 171–178. Source: assets/cortex_glasser.json.

Subcortex

19 deep-brain structures (thalamus, hippocampus, amygdala, basal-ganglia complex, brainstem nuclei, cerebellum) rendered as discrete clickable meshes; each pins a causal chain. Source: assets/subcortex.json.

Connectome

White-matter edge sample drawn between parcel centroids, following the convention of the diffusion-MRI tractography literature. Source: assets/connectome_edges.json.

EEG bands

Canonical band-power traces (δ, θ, α, β, γ) drive the background phase + amplitude of the cortical wave layer. Source: assets/eeg_band_powers.json.

Worden RE trace

Pre-computed baseline of the Requirement Equation against free energy under the joint update rule, used to seed ω_W slider initialisation. Source: assets/traces/worden_re_baseline.json.

§ 03 · The falsifiable core (PWTC)

Worden's Projective Wave Theory of Consciousness — the five equations under test

These are the five equations whose survival of contact with real neural recordings determines whether PWTC stands. They are tagged PWTC throughout the side panel of the viewer.

P1 · Ⅰ
PWTC

Wave-weighted state transition (wave + state substrate)

$$P(s' \,|\, s, a) \;=\; \frac{\sum_k |\psi_k(s, a)|^2 \; T_k(s' \,|\, s, a)}{\sum_k |\psi_k(s, a)|^2}$$

Worden, R. (2026), The Projective Wave Theory of Consciousness, Frontiers in Psychology (10.3389/fpsyg.2026.1674983; preprint arXiv:2405.12071). Neural activity couples to a wave excitation that holds the analogue model of space. Discrete state s and wave amplitudes ψ_k co-evolve through a cycle-scheduled projection/collapse.

P11 · Ⅺ
PWTC
LIVE

Worden Requirement Equation + joint Friston/Worden update

$$R(s) \;=\; \sum_{i=1}^{N} w_i\,\bigl|\,g_i - p_i(s)\,\bigr| \\[6pt] \theta_{t+1} \;=\; \theta_t \,-\, (1-\omega_W)\,\eta\,\nabla_{\!\theta} F \,+\, \omega_W\,\eta\,\nabla_{\!\theta} E$$

Worden, R. (2024), The Requirement for Cognition, in an Equation, arXiv:2405.08601. Goals g_i are the system's requirements; p_i(s) its current performance; R(s) is minimised by adaptive behaviour. Runs as a live joint update beside Friston's free-energy gradient — the ω_W slider sets survival vs. fitness.

P12 · Ⅻ
PWTC

Projective wave in the thalamus + decoding

$$\psi(\mathbf{k},t) = \sum_{n=1}^{N} a_n\,\exp\!\bigl(i\,(\mathbf{k}_n \cdot \rho(\mathbf{r}) - \omega(\mathbf{k}_n)\,t)\bigr) \\[4pt] I(\mathbf{r}) \;=\; \bigl|\,\mathcal{F}^{-1}\{\psi\}(\mathbf{r})\,\bigr|^2$$

Worden, R. (2026), PWTC, Frontiers / arXiv:2405.12071. The thalamus holds an analogue model of 3-D space as a wave whose components have wave-vectors k corresponding to projectively-transformed positions ρ(r). The spatial form of conscious experience is the inverse Fourier transform of that wave.

P15 · XV
PWTC

Perspectival imagination (wave transform under SE(3))

$$\rho:\ \mathbb{R}^3 \to \mathbb{P}^3 \quad\text{(projective)} \\[4pt] \psi(\mathbf{k},t)=\sum_n a_n\,e^{\,i(\mathbf{k}_n\cdot\rho(\mathbf r)-\omega_n t)} \\[4pt] \text{imagine: } \rho \mapsto \rho\circ g,\ \ g\in SE(3) \;\Rightarrow\; \text{lines}\to\text{lines}$$

Worden, R. (2024) Spatial Cognition: A Wave Hypothesis, arXiv:2405.10112. Imagination applies an imagined viewpoint change g to the thalamic wave so anticipation and planning run in the same projective space as perception. Projective invariance — straight world-lines map to straight image-lines — is Worden's reason the internal model must be projective rather than arbitrary.

P16 · XVI
PWTC

Cortical microcircuit — Van der Pol + Kuramoto neurons reading/writing the wave

$$x_i(t) = A_i(t)\,\cos\theta_i(t) \\[4pt] \dot\theta_i = \omega_i + \tfrac{K}{N}\sum_j \sin(\theta_j-\theta_i), \quad \dot A_i = \mu_i A_i\bigl(1-A_i^{2}\bigr) \\[4pt] \text{spike}_i \;\Leftrightarrow\; x_i \uparrow \theta_{\mathrm{thr}}$$

Worden, R. (2026) PWTC; single-neuron membranes as Van der Pol relaxation oscillators — Van der Pol & Van der Mark (1928); FitzHugh (1961); Nagumo (1962). Populations synchronise via Kuramoto coupling with order parameter R=|⟨e^iθ⟩|. Cortex is ~80% excitatory pyramidal / ~20% inhibitory — DeFelipe & Fariñas (1992); Markram et al. (2004). Dale's principle — Dale (1935); Eccles (1976). Gamma rhythms — Buzsáki & Wang (2012). Precision-from-phase — Feldman & Friston (2010).

Worden's claim is not that the cortex is the seat of consciousness; it is that the thalamic projective wave is, and the cortex is its sensor and actuator. The viewer makes this geometry literal: when the wave layer is active, the cortical mesh reads from and writes to it, and the perceptual decode is rendered as a separate field.

§ 04 · Mechanistic scaffold — 19 layer equations

The established neuroscience PWTC must interoperate with

Each row is one toggleable layer in the side panel. The LIVE tag marks layers whose update() actually steps the ODE / wave / accumulator each frame and writes to ctx.state; the others render an anatomical pulse driven by a precomputed trace. Layers tagged PWTC belong to Worden's core (§ 03 above, restated here in their full panel position).

P2 · Ⅱ

Phase coherence

$$R \;=\; \Bigl|\,\langle e^{i\theta}\rangle\,\Bigr| \qquad \pi_i \;=\; \pi_{\text{base}} + \alpha \cos\theta_i$$

Kuramoto (1975) order parameter; Friston & Feldman (2010) precision-weighted attention. Used in Worden's framework as the substrate over which the thalamic wave couples to neurons.

P3 · Ⅲ

Variational free energy (FEP)

$$F \;=\; \mathbb{E}_{q}\!\bigl[\log q(z) - \log p(o,z)\bigr] \;=\; \underbrace{\mathbb{E}_q[-\log p(o\,|\,z)]}_{\text{accuracy}} \,+\, \underbrace{\mathrm{KL}\!\bigl(q(z)\,\|\,p(z)\bigr)}_{\text{complexity}}$$

Friston (2010) Free Energy Principle. In PWTC, free-energy minimisation is what the cortex uses to maintain coupling with the thalamic wave.

P4 · Ⅳ

Kuramoto coupling with adaptive K

$$\frac{d\theta_i}{dt} \;=\; \omega_i \,+\, \frac{K}{N}\sum_{j=1}^{N}\sin(\theta_j-\theta_i) \qquad K_{t+1} = K_t + \eta\,\nabla_{\!K} F$$

Kuramoto (1975). Free-energy gradient adapts coupling K live. In Worden's framework, Kuramoto coupling maintains the thalamic wave but is not itself the seat of consciousness.

P5 · Ⅴ

Predictive-coding hierarchy

$$\varepsilon_\ell \;=\; \pi_\ell\bigl(\mu_\ell - g(\mu_{\ell+1})\bigr) \qquad \dot{\mu}_\ell \;=\; -\,\frac{\partial F}{\partial \mu_\ell}$$

Rao & Ballard (1999); Friston (2005). Bidirectional predictive coding. Worden argues these predictions are not the conscious content — they are how neurons read from and write to the projective wave.

P6 · Ⅵ

Hopfield CA3 (hippocampal autoassociator)

$$W_{ij} \;=\; \sum_{\mu=1}^{P} \xi_i^{\mu}\,\xi_j^{\mu} \qquad s_i(t+1) \;=\; \operatorname{sign}\!\Bigl(\sum_j W_{ij}\,s_j(t)\Bigr)$$

Hopfield (1982). CA3 content-addressable attractor; stores the priors that condition the wave's Bayesian update over remembered spatial scenes.

P7 · Ⅶ

HPA-axis distress (CRH → ACTH → cortisol negative feedback)

$$\frac{dC}{dt} \;=\; k\,A \,-\, w\,C \qquad \frac{dA}{dt} \;=\; \frac{p}{1+(C/K_d)^{\,n}} \,-\, q\,A$$

Glucocorticoid feedback — Sapolsky, Romero & Munck (2000), Endocrine Reviews. Amygdala-gated stress drive. The affective substrate that gates which sense-data the thalamic wave incorporates.

P8 · Ⅷ

Somatic markers (insular / vmPFC)

$$c_i \leftarrow c_i \cdot \bigl(1 - \omega \cdot \hat{m}_i^{\,-}\bigr) \qquad \hat{m}_i^{\,-} \;=\; \max\!\bigl(0,\, -\mathrm{soma}(\text{state})\bigr)$$

Damasio (1994) Somatic Marker Hypothesis. Negative mood deflates epistemic confidence; in PWTC, this is how the body signals which sense-data should be prioritised in the wave's Bayesian update.

P9 · Ⅸ

Thalamocortical ignition (global workspace)

$$i^* = \operatorname{argmax}_i\bigl(\pi_i \cdot a_i\bigr) \qquad a_{i^*} \leftarrow 1.5\,a_{i^*}, \;\; a_{j\neq i^*} \leftarrow 0.5\,a_j$$

Baars (1988); Dehaene & Naccache (2001) global workspace ignition. Worden places the thalamus at the centre of the ignition pattern: ignition is the thalamic wave being driven into coherent excitation by aligned neural input.

P10 · Ⅹ

aPFC metacognition

$$C_{\text{meta}} \;=\; \frac{1}{1 + |F|}\sum_{i=1}^{8} w_i\,c_i \qquad \eta_{\text{balance}} \;=\; \frac{\text{coherence}}{1 + |F|}$$

Fleming & Lau (2014) anterior PFC metacognitive arbitration. The metacognitive estimate gates how readily cortex commits to the current wave-derived spatial percept.

P13 · XIII
LIVE

Homeostatic divergence (allostasis)

$$\frac{db}{dt} \;=\; \alpha\,\nabla_{\!b} F(b,o) \,+\, \beta\,\bigl(o_t - g(b)\bigr) \,+\, \lambda\,\nabla_{\!b} H_D\!\bigl(q \,\|\, q^*\bigr)$$

H_D = KL(q ‖ q*) is the divergence from a low-entropy preferred distribution. Allostasis — the brain defends specific physiological set-points, not merely low surprise — Sterling & Eyer (1988); Sterling (2012).

P14 · XIV

Communication policy (active-inference over speech acts)

$$G(\pi) \;=\; \mathbb{E}_{q}\!\bigl[\ln q(o,\theta\,|\,\pi) - \ln p(o,\theta\,|\,C)\bigr] \\[4pt] \pi^* \;=\; \operatorname{softmax}\!\Bigl(-\,\tfrac{G(\pi)}{\tau}\Bigr), \quad \pi \in \{\text{withhold}, \text{qualify}, \text{meta}, \text{speak}\}$$

Active-inference policy selection — Friston et al. (2017). Silence is a first-class outcome. Built on Worden's Requirement Equation: the system speaks only when speaking reduces R for the receiver.

P17 · XVII

Cortical travelling waves

$$\phi(\mathbf r,t)=\mathbf k\cdot\mathbf r-\omega t,\quad v_\phi=\omega/|\mathbf k|\approx 0.1\text{–}0.8\,\mathrm{m/s} \\[4pt] a(\mathbf r,t)=\sum_{m} A_m\cos\!\bigl(\mathbf k_m\cdot\mathbf r-\omega_m t\bigr)$$

Muller, Chavane, Reynolds & Sejnowski (2018), Nat. Rev. Neurosci. Waves sweep cortex at 0.1–0.8 m/s during perception, memory, and sleep. In PWTC the cortical wave reads the thalamic projective wave.

P18 · XVIII

Sleep & dreaming (sharp-wave-ripple replay)

$$\text{UP/DOWN slow oscillation} \sim 0.8\,\mathrm{Hz} \ \text{gates sharp-wave ripples} \\[4pt] \text{hippocampus} \rightarrow \text{neocortex replay (consolidation)}$$

Wilson & McNaughton (1994); Buzsáki (2015); Diekelmann & Born (2010). Offline, the projective wave runs on remembered scenes — a dream.

P19 · XIX
LIVE

Drift-diffusion + sequential Bayesian belief update

$$\dot x \;=\; v(t)\,dt + \sigma\,dW, \qquad \text{decide when } |x|\ge a \\[4pt] q_{t+1}(s) \;\propto\; q_{t}(s)\,p(o_{t+1}\,|\,s)\,p(s)$$

Ratcliff (1978), Psychological Review drift-diffusion model; Mathys et al. (2011) hierarchical Gaussian filter; Gold & Shadlen (2007) LIP/FEF/DLPFC correlates. Steps the SDE each frame (Wiener increment σ√dt·z), reads drift from ctx.state.pred_error, and applies a Bayesian update to a four-bin posterior q(s) on each boundary crossing. Closes the largest gap in the perception chain — there is now an explicit accumulating belief variable behind the wave decode.

§ 05 · Six causal chains (pinned by clicking the structure)

Each chain walks four anatomical stages from raw input to emergent expression

Click the named structure in the viewer to pin the chain in the side panel. Each stage names the anatomical site, the equation that runs at that hop, and the citation that grounds it.

Chain 1 · Conscious perception

pinned by clicking THALAMUS

V1 · primary sensory cortex

$p(o,s)=p(o\|s)\,p(s)$

Generative model — Friston (2010) FEP.

Association cortex

$F=\mathbb E_q[\log q(s)-\log p(o,s)]$

Variational free energy — Friston (2010).

Thalamus · PWTC

$\psi(\mathbf k,t)=\sum_n a_n e^{i(\mathbf k_n\cdot\rho(\mathbf r)-\omega_n t)}$

Worden PWTC, arXiv:2405.12071 §6. LGN, MGN, VPL/VPM host the projective wave.

Global access · percept

$I(\mathbf r)=|\mathcal F^{-1}\{\psi\}(\mathbf r)|^2$

Decoding rule — Worden; broadcast — Dehaene & Changeux (2011).

Chain 2 · Episodic memory

pinned by clicking HIPPOCAMPUS

Neocortex · experience pattern

$\xi^\mu\in\{-1,+1\}^N$

Distributed activity pattern, conveyed via entorhinal cortex.

CA3 · autoassociator

$W_{ij}=\sum_\mu \xi_i^\mu \xi_j^\mu$

Treves & Rolls (1994); Hopfield (1982).

SWR reinstatement

$\text{SWR: hippocampus}\to\text{neocortex}$

Sharp-wave-ripple replay — Buzsáki (2015).

Recollection

$\hat\xi\approx\xi^\mu$

Pattern completion → conscious recollection.

Chain 3 · Affective salience

pinned by clicking AMYGDALA

Thalamic low road

$o_t\to\text{amygdala (subcortical)}$

Subcortical fast route — LeDoux (1996).

Amygdala · aversive learning

$\Delta V=\alpha\beta(\lambda-\sum V)$

Rescorla–Wagner associative value; LeDoux (2000).

Salience network · precision

$\pi_i\uparrow\Rightarrow \varepsilon_i=\pi_i(\mu_i-g(\mu_{i+1}))$

Anterior insula / dACC — Menon & Uddin (2010).

Autonomic + felt response

$\text{CeA}\to\text{hypothalamus / brainstem}$

Somatic-marker — Damasio (1996).

Chain 4 · Action selection

pinned by clicking BASAL GANGLIA

Cortex · candidate actions

$\{a_1,a_2,\dots,a_n\}$

Corticostriatal motor plans (M1, premotor, PFC).

Striatum · DA-gated

$\delta_t=r_t+\gamma V(s_{t+1})-V(s_t)$

DA reward-prediction error — Schultz (1997). D1/D2 Go/NoGo.

Pallidal output · disinhibition

$i^*=\operatorname{argmax}_i(\pi_i a_i)$

GPi/SNr — Mink (1996); Redgrave (1999).

Motor thalamus → M1

$u_t=a_{i^*}$

Released as movement.

Chain 5 · Arousal & precision

pinned by clicking BRAINSTEM

Reticular activating system

$a(t)=\text{reticular arousal}$

ARAS — Moruzzi & Magoun (1949).

Neuromodulatory nuclei · gain

$\pi=\pi_0 e^{\gamma a}$

LC gain — Aston-Jones & Cohen (2005); NA, 5-HT, DA, ACh.

Thalamocortical modulation

$K,\pi \propto a$

Intralaminar thalamic nuclei — Steriade (2000).

Global brain state

$\text{wake}\leftrightarrow\text{NREM}\leftrightarrow\text{REM}$

Sleep-state switch — Saper et al. (2005); VLPO, orexin.

Chain 6 · Forward model

pinned by clicking CEREBELLUM

Motor cortex · efference copy

$u_t \text{ (command + efference copy)}$

Efference copy — von Holst & Mittelstaedt (1950).

Cerebellum · forward model

$\hat o_{t+1}=f_w(x_t,u_t)$

Internal forward model — Wolpert (1998); Ito (2008). PF–Purkinje LTD.

Cerebello-thalamo-cortical loop

$\Delta u = -\eta(o-\hat o)$

Deep cerebellar nuclei → ventrolateral thalamus → cortex.

Coordinated action

$\text{smooth, well-timed behaviour}$

Emergent motor coordination.

§ 06 · Transparency

Live integration vs. display-only

Each layer tile in the viewer carries a LIVE or DISPLAY badge. The distinction is structural, not cosmetic:

Live integrated

The layer's update() steps the ODE / wave / stochastic accumulator each frame and writes new state into ctx.state that other layers can read. P11 (Requirement Equation joint update), P13 (homeostatic divergence), and P19 (drift-diffusion + sequential Bayes) are live. P11's ω_W slider is wired to the integrator and changes the trajectory in real time.

Display

The layer renders an anatomically-correct pulse driven by a pre-computed trace or by other live layers it reads. The equation shown is the source equation, not a runtime simulation. This is the honest default for layers where the full simulation is computationally infeasible in the browser (e.g. P12's full thalamic wave field) or where the equation is a literature scaffold rather than novel-theory core.

The badge exists so a reader cannot mistake a literature-reference equation for evidence that the equation has been simulated.

§ 07 · Methods

How the page is rendered and what it cites

Render stack

Three.js (WebGL2) for the cortex + subcortex meshes; KaTeX 0.16.11 for in-panel equation rendering; vanilla ES modules for the layer / chain definitions. No build step.

Primary citations

Worden (2024) arXiv:2405.08601, arXiv:2405.10112, arXiv:2405.12071; Worden (2026) Frontiers in Psychology 10.3389/fpsyg.2026.1674983; Friston (2010); Glasser et al. (2016); Hopfield (1982); Ratcliff (1978); Buzsáki (2015); Muller et al. (2018). Full citation list inline in §§ 03–05 above.

§ 08 · Acknowledgment

On whose shoulders this stands

This page rests on the work of others. The Projective Wave Theory of Consciousness and the Requirement Equation are Robert Worden's; the Free Energy Principle and the active-inference framework are Karl Friston's. I am deeply grateful to both — and to Robert Worden in particular for the generosity of putting forward a novel, falsifiable theory of consciousness in a form that other people can pick up, render, integrate, and test. The synthesis itself is mine: the choices of which equations to include, which to omit, which to integrate live and which to render as anatomical illustration, and how to wire them together into a single inspectable surface. Any error of attribution, of interpretation, of implementation, or of judgment that the reader finds here is mine alone.