wifi-densepose/examples/exo-ai-2025/research/03-time-crystal-cognition/mathematical_framework.md

Mathematical Framework: Floquet Theory for Cognitive Time Crystals

1. Floquet Formalism for Neural Dynamics

1.1 Continuous-Time Neural Field Equations

Consider a population of N neurons with firing rates \mathbf{r}(t) = [r_1(t), ..., r_N(t)]^T:

\tau \frac{d\mathbf{r}}{dt} = -\mathbf{r} + f(W\mathbf{r} + \mathbf{I}(t)) + \boldsymbol{\eta}(t)

where:

• \tau : neural time constant
• W : synaptic connectivity matrix (asymmetric: W_{ij} \neq W_{ji})
• f(\cdot) : activation function (e.g., \tanh, sigmoid)
• \mathbf{I}(t) = \mathbf{I}(t + T) : periodic external input (driving force)
• \boldsymbol{\eta}(t) : Gaussian white noise with \langle \eta_i(t) \eta_j(t') \rangle = 2D\delta_{ij}\delta(t-t')

1.2 Floquet Decomposition

For periodic systems, the general solution can be written as:

\mathbf{r}(t) = \sum_{\alpha} c_{\alpha} e^{\mu_{\alpha} t} \mathbf{u}_{\alpha}(t)

where:

• \mathbf{u}_{\alpha}(t + T) = \mathbf{u}_{\alpha}(t) : Floquet modes (periodic)
• \mu_{\alpha} : Floquet exponents (complex)
• c_{\alpha} : coefficients determined by initial conditions

1.3 Floquet Multipliers

The Floquet multipliers \lambda_{\alpha} relate to exponents via:

\lambda_{\alpha} = e^{\mu_{\alpha} T}

Stability conditions:

• |\lambda_{\alpha}| < 1 : stable
• |\lambda_{\alpha}| = 1 : marginal (limit cycle)
• |\lambda_{\alpha}| > 1 : unstable

1.4 Subharmonic Response Criterion

Definition: System exhibits subharmonic response of order k if:

\mathbf{r}(t + kT) = \mathbf{r}(t) \quad \text{but} \quad \mathbf{r}(t + T) \neq \mathbf{r}(t)

Floquet criterion: Exists Floquet exponent with

\mu_{\alpha} = i\frac{2\pi m}{kT} \quad \text{for integers } m, k \text{ with } \gcd(m,k)=1

For period-doubling (k=2):

\mu = i\frac{\pi}{T} \implies \lambda = e^{i\pi} = -1

Interpretation: After one period T, state vector is negated; after two periods 2T, it returns to original.

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2. Time Crystal Order Parameter

2.1 Definition

The time crystal order parameter for subharmonic order k is:

M_k(t) = \frac{1}{N} \left| \sum_{i=1}^{N} e^{i k \omega_0 \phi_i(t)} \right|

where:

• \omega_0 = 2\pi/T : driving frequency
• \phi_i(t) : phase of neuron i relative to driving force
• k : subharmonic order (typically 2 for period-doubling)

2.2 Phase Extraction

From firing rate r_i(t), extract instantaneous phase via Hilbert transform:

\tilde{r}_i(t) = r_i(t) + i \mathcal{H}[r_i](t) \phi_i(t) = \arg(\tilde{r}_i(t))

where \mathcal{H} is the Hilbert transform.

2.3 Time-Averaged Order Parameter

For stationary dynamics:

\bar{M}_k = \lim_{T_{\text{avg}} \to \infty} \frac{1}{T_{\text{avg}}} \int_0^{T_{\text{avg}}} M_k(t) dt

CTC phase: \bar{M}_k > M_{\text{crit}} (typically M_{\text{crit}} \sim 0.5)

Non-CTC phase: \bar{M}_k \approx 0

2.4 Temporal Correlation Function

C_k(\tau) = \langle M_k(t) M_k^*(t + \tau) \rangle_t

CTC signature: Long-range correlations

• Power-law decay: C_k(\tau) \sim \tau^{-\alpha} with 0 < \alpha < 1
• Or persistent: C_k(\tau) \to C_{\infty} > 0

Non-CTC: Exponential decay C_k(\tau) \sim e^{-\tau/\tau_c}

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3. Effective Hamiltonian and Energy Landscape

3.1 Neural Energy Function

Define Lyapunov function (energy) for symmetric component of W:

E(\mathbf{r}) = -\frac{1}{2} \mathbf{r}^T W_S \mathbf{r} - \mathbf{b}^T \mathbf{r} + \sum_i \int_0^{r_i} f^{-1}(x) dx

where W_S = (W + W^T)/2 is symmetric part.

For asymmetric W:

W = W_S + W_A

where W_A = (W - W^T)/2 is antisymmetric.

3.2 Gradient and Circulatory Dynamics

Dynamics decompose into:

\frac{d\mathbf{r}}{dt} = -\nabla E(\mathbf{r}) + W_A \mathbf{r}

• Gradient term: -\nabla E drives toward minima
• Circulatory term: W_A \mathbf{r} causes rotation in state space

Key insight: W_A \neq 0 (asymmetric connectivity) enables limit cycles and time crystals.

3.3 Floquet Effective Hamiltonian

For periodically driven system, define effective Hamiltonian via Magnus expansion:

H_{\text{eff}} = H_0 + \sum_{n=1}^{\infty} H_{\text{eff}}^{(n)}

where:

H_0 = \frac{1}{T} \int_0^T H(t) dt H_{\text{eff}}^{(1)} = \frac{1}{2T} \int_0^T dt_1 \int_0^{t_1} dt_2 \, [H(t_1), H(t_2)]

Higher orders involve nested commutators.

3.4 High-Frequency Limit

For \omega_0 \tau \gg 1 (high-frequency driving relative to neural timescale):

H_{\text{eff}} \approx H_0 + \mathcal{O}(1/\omega_0)

System approximately described by time-averaged Hamiltonian.

CTC emergence: Corrections H_{\text{eff}}^{(n)} create frequency-dependent interactions enabling subharmonic responses.

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4. Prethermal Dynamics and Heating

4.1 Heating Rate

Floquet systems generally absorb energy and heat to infinite temperature. Heating rate:

\frac{dE}{dt} = \gamma E

where \gamma depends on drive frequency and system properties.

4.2 Prethermal Regime

For sufficiently fast driving (\omega_0 \gg \omega_{\text{local}}):

t_{\text{pretherm}} \sim \frac{1}{\gamma} e^{c \omega_0/\omega_{\text{local}}}

where c is a constant, \omega_{\text{local}} is local energy scale.

CTC lifetime: Must have t_{\text{WM}} \ll t_{\text{pretherm}}

For theta oscillations (8 Hz) and neural timescales (100 Hz):

\omega_0/\omega_{\text{local}} \sim 0.08 \implies t_{\text{pretherm}} \sim e^{0.08c}

With appropriate parameters, t_{\text{pretherm}} can be seconds to minutes.

4.3 Dissipation and Stabilization

Including dissipation via coupling to heat bath:

\tau \frac{d\mathbf{r}}{dt} = -\mathbf{r} + f(W\mathbf{r} + \mathbf{I}(t)) - \gamma_D (\mathbf{r} - \mathbf{r}_{\text{rest}}) + \boldsymbol{\eta}(t)

Dissipation term -\gamma_D (\mathbf{r} - \mathbf{r}_{\text{rest}}) removes energy, preventing heating.

Balance condition for stable CTC:

\text{Energy input from drive} = \text{Energy dissipation}

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5. Phase Diagram and Bifurcations

5.1 Control Parameters

• A : drive amplitude
• \omega_0 : drive frequency
• J : coupling strength (connectivity magnitude)
• N : system size (number of neurons)

5.2 Period-Doubling Bifurcation

Consider parameterizing by drive amplitude A:

Subcritical regime (A < A_c):

• System oscillates at drive frequency \omega_0
• Stable fixed point in Poincaré section

Supercritical regime (A > A_c):

• Period-doubling: oscillation at \omega_0/2
• Stable limit cycle with period 2T

Critical amplitude:

A_c \propto \frac{1}{\sqrt{N}} \cdot \frac{\omega_0}{\gamma_D}

Scaling: A_c decreases with system size N (easier to form CTC in larger systems).

5.3 Phase Diagram

In (A, \omega_0) space:

ω₀
 │
 │     Heating Regime
 │   (Too fast, no CTC)
 │─────────────────────
 │         │
 │  Non-   │   CTC
 │  CTC    │  Regime
 │         │   k=2
 │         │
 │─────────┴──────────
 │    Quasistatic
 │   (Too slow)
 └─────────────────── A
        Ac

5.4 Higher-Order Subharmonics

Increasing A further can lead to:

• k=2 : period-doubling
• k=3 : period-tripling
• k=4 : period-quadrupling
• ...
• Chaos: Route to chaos via period-doubling cascade

Feigenbaum constant: Ratio of successive bifurcation points converges to \delta \approx 4.669

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6. Many-Body Effects and Localization

6.1 Mean-Field Approximation

For large N, treat average field:

m(t) = \frac{1}{N} \sum_i r_i(t)

Single neuron dynamics:

\tau \frac{dr_i}{dt} = -r_i + f(Jm(t) + I_i(t) + h_i) + \eta_i(t)

where h_i is random field (quenched disorder).

6.2 Self-Consistency Equation

m(t) = \int dh \, P(h) \, \langle r(t; h) \rangle

where P(h) is distribution of disorder, \langle \cdot \rangle averages over noise.

6.3 Localization and DTC Stability

Disorder: Heterogeneous synaptic weights, thresholds create "localization"

Effective localization length:

\xi \sim \frac{J^2}{\sigma_h^2}

where \sigma_h^2 is variance of disorder.

CTC stability: Requires sufficient disorder (\sigma_h large) to prevent ergodic exploration of state space.

Critical disorder:

\sigma_h > \sigma_c \sim J \sqrt{N}

Below this, system thermalizes and CTC collapses.

6.4 Many-Body Localization Analogue

In quantum DTCs, MBL prevents thermalization. In cognitive systems:

"Synaptic localization": High-dimensional, disordered connectivity landscape creates local minima trapping activity patterns.

Criterion: Decay of correlations in connectivity: \langle W_{ij} W_{kl} \rangle \sim \delta_{ik}\delta_{jl} (sparse, random connectivity)

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7. Spectral Analysis and Detection

7.1 Power Spectral Density

For time series r(t), compute PSD via Fourier transform:

S(\omega) = \lim_{T \to \infty} \frac{1}{T} \left| \int_0^T r(t) e^{-i\omega t} dt \right|^2

CTC signature:

• Peak at drive frequency: S(\omega_0)
• Peak at subharmonic: S(\omega_0/k)
• Ratio: R_k = S(\omega_0/k) / S(\omega_0)

Detection criterion: R_k > R_{\text{thresh}} (e.g., R_{\text{thresh}} = 1 for period-doubling)

7.2 Cross-Frequency Coupling

Between regions A and B:

C_{AB}(\omega_1, \omega_2) = \left| \langle r_A(\omega_1) r_B^*(\omega_2) \rangle \right|

CTC prediction:

• Strong coupling at (\omega_0, \omega_0/k): Region A at drive frequency, region B at subharmonic
• Or vice versa
• Indicates coordinated time crystal dynamics across brain regions

7.3 Bicoherence

Measure phase-coupling:

b(\omega_1, \omega_2) = \frac{\left| \langle r(\omega_1) r(\omega_2) r^*(\omega_1 + \omega_2) \rangle \right|}{\sqrt{\langle |r(\omega_1) r(\omega_2)|^2 \rangle \langle |r(\omega_1+\omega_2)|^2 \rangle}}

CTC signature: Peak at (\omega_0/k, \omega_0/k) indicating frequency-mixing that generates subharmonic.

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8. Stochastic Floquet Theory

8.1 Fokker-Planck Equation

For probability density P(\mathbf{r}, t):

\frac{\partial P}{\partial t} = -\sum_i \frac{\partial}{\partial r_i} [F_i(\mathbf{r}, t) P] + D \sum_i \frac{\partial^2 P}{\partial r_i^2}

where F_i(\mathbf{r}, t) = \frac{1}{\tau}[-r_i + f_i(W\mathbf{r} + \mathbf{I}(t))] is drift.

8.2 Floquet-Fokker-Planck

Decompose P into Floquet modes:

P(\mathbf{r}, t) = \sum_{\alpha} e^{\lambda_{\alpha} t} P_{\alpha}(\mathbf{r}, t)

where P_{\alpha}(\mathbf{r}, t + T) = P_{\alpha}(\mathbf{r}, t).

8.3 Leading Eigenvalue

Long-time behavior dominated by leading eigenvalue \lambda_0:

P(\mathbf{r}, t) \xrightarrow{t \to \infty} e^{\lambda_0 t} P_0(\mathbf{r}, t)

Stationary CTC: \lambda_0 = 0 with periodic P_0(\mathbf{r}, t) exhibiting subharmonic structure.

8.4 Noise-Induced Transitions

Critical noise level D_c:

• D < D_c: CTC phase stable
• D > D_c: CTC collapses to non-CTC

D_c \propto (A - A_c)^{\gamma}

where \gamma \sim 2 near bifurcation (mean-field exponent).

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9. Finite-Size Scaling

9.1 Scaling Hypothesis

Near phase transition, observables obey scaling laws:

M_k = N^{-\beta/\nu} \tilde{M}(N^{1/\nu}(A - A_c))

where \beta, \nu are critical exponents, \tilde{M} is scaling function.

9.2 Correlation Length

Spatial correlations:

\langle r_i r_j \rangle \sim e^{-|i-j|/\xi}

Divergence at critical point:

\xi \sim |A - A_c|^{-\nu}

9.3 Prethermal Lifetime Scaling

t_{\text{pretherm}} \sim N^{\alpha} e^{\beta N}

Exponential scaling: Prethermal lifetime increases exponentially with system size.

For neural populations:

• Small network (N \sim 100): t_{\text{pretherm}} \sim milliseconds
• Large network (N \sim 10^6): t_{\text{pretherm}} \sim seconds to minutes

This explains why working memory in large-scale cortical networks can persist for seconds.

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10. Information-Theoretic Measures

10.1 Temporal Mutual Information

Between time slices separated by \tau:

I(\tau) = \sum_{r(t), r(t+\tau)} P(r(t), r(t+\tau)) \log \frac{P(r(t), r(t+\tau))}{P(r(t))P(r(t+\tau))}

CTC signature: Peaks at \tau = kT indicating long-range temporal correlations at subharmonic period.

10.2 Integrated Information (Φ)

For partition \mathcal{P} of system:

\Phi = \min_{\mathcal{P}} \, \text{EMD}(P_{\text{whole}}, P_{\text{part}})

where EMD is earth mover's distance between distributions.

Hypothesis: CTC phase has higher \Phi than non-CTC due to:

• Long-range temporal correlations
• Many-body nature (cannot decompose into independent parts)

10.3 Entropy Production Rate

Nonequilibrium measure:

\dot{S} = -\int d\mathbf{r} \, P(\mathbf{r}, t) \nabla \cdot \mathbf{F}(\mathbf{r}, t)

CTC signature: Positive steady-state entropy production \langle \dot{S} \rangle > 0, confirming nonequilibrium nature.

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11. Perturbation Response Theory

11.1 Linear Response

Apply small perturbation \delta I(t):

\delta r_i(t) = \int_{-\infty}^{t} \chi_{ij}(t, t') \delta I_j(t') dt'

where \chi_{ij}(t, t') = \chi_{ij}(t + T, t' + T) is Floquet response function.

11.2 Floquet Susceptibility

Fourier transform:

\chi_{ij}(\omega) = \sum_n \chi_{ij}^{(n)}(\omega) e^{in\omega_0 t}

CTC signature: Resonances at \omega = \omega_0/k indicating enhanced response at subharmonic.

11.3 Phase Response Curve (PRC)

For perturbation at phase \phi of limit cycle:

\Delta \phi = Z(\phi) \cdot \delta I

where Z(\phi) is phase response curve.

CTC property: PRC has period kT (not T), reflecting subharmonic structure.

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12. Numerical Methods

12.1 Direct Simulation

Euler-Maruyama scheme for stochastic dynamics:

r_i(t + \Delta t) = r_i(t) + \frac{\Delta t}{\tau}[-r_i(t) + f_i(W\mathbf{r}(t) + \mathbf{I}(t))] + \sqrt{2D\Delta t} \, \xi_i

where \xi_i \sim \mathcal{N}(0, 1).

12.2 Floquet Analysis via Monodromy Matrix

1. Solve one period from initial condition \mathbf{r}_0
2. Obtain \mathbf{r}(T)
3. Repeat for N initial conditions forming basis
4. Construct monodromy matrix M with columns \mathbf{r}^{(i)}(T)
5. Eigenvalues of M are Floquet multipliers \lambda_{\alpha}

12.3 Order Parameter Computation

def compute_order_parameter(firing_rates, drive_frequency, k=2):
    """
    Compute time crystal order parameter M_k

    Args:
        firing_rates: array of shape (N_neurons, N_timesteps)
        drive_frequency: driving frequency (Hz)
        k: subharmonic order

    Returns:
        M_k: order parameter as function of time
    """
    from scipy.signal import hilbert

    # Extract phases via Hilbert transform
    analytic_signal = hilbert(firing_rates, axis=1)
    phases = np.angle(analytic_signal)

    # Compute order parameter
    omega_0 = 2 * np.pi * drive_frequency
    M_k = np.abs(np.mean(np.exp(1j * k * omega_0 * phases), axis=0))

    return M_k

# Time average
M_k_avg = np.mean(M_k)

12.4 Spectral Analysis

from scipy import signal

def detect_subharmonics(firing_rate, dt, drive_freq, k_max=4):
    """
    Detect subharmonic peaks in power spectrum

    Returns:
        subharmonic_ratios: Power(f/k) / Power(f) for k=2,3,4,...
    """
    freqs, psd = signal.welch(firing_rate, fs=1/dt)

    ratios = {}
    drive_idx = np.argmin(np.abs(freqs - drive_freq))

    for k in range(2, k_max+1):
        subharmonic_freq = drive_freq / k
        sub_idx = np.argmin(np.abs(freqs - subharmonic_freq))
        ratios[k] = psd[sub_idx] / psd[drive_idx]

    return ratios

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13. Connection to Experimental Observables

13.1 EEG/MEG Power Spectrum

Measured: Voltage fluctuations V(t) at scalp

Model: V(t) \propto \sum_i r_i(t) w_i where w_i are spatial weights

CTC prediction:

• Peak at theta frequency (~8 Hz) from drive
• Peak at alpha frequency (~4 Hz = theta/2) from period-doubling

Test: Ratio R_2 = P_{\alpha}/P_{\theta} increases during working memory maintenance

13.2 Single-Neuron Recordings

Measured: Spike trains \{t_i^{(n)}\}\_{n=1}^{N_{\text{spikes}}} for neuron i

Model: Firing rate r_i(t) determines spike probability

CTC prediction:

• Inter-spike intervals cluster at multiples of T/k
• Phase-locking to subharmonic of LFP

13.3 Functional Connectivity

Measured: Correlation C_{ij} = \langle r_i(t) r_j(t) \rangle between regions i, j

CTC prediction:

• Frequency-specific connectivity at f/k
• Increase in connectivity during CTC phase vs. baseline

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14. Summary of Key Equations

Concept	Equation	Description
Neural dynamics	\tau \frac{d\mathbf{r}}{dt} = -\mathbf{r} + f(W\mathbf{r} + \mathbf{I}(t))	Periodically driven neural field
Floquet decomposition	\mathbf{r}(t) = \sum_{\alpha} c_{\alpha} e^{\mu_{\alpha} t} \mathbf{u}_{\alpha}(t)	General solution
Period-doubling	\mu = i\pi/T \implies \lambda = -1	Floquet multiplier for k=2
Order parameter	$M_k = \frac{1}{N}\left	\sum_i e^{ik\omega_0\phi_i}\right
Critical amplitude	A_c \propto \frac{1}{\sqrt{N}} \frac{\omega_0}{\gamma_D}	Bifurcation point
Prethermal time	t_{\text{pretherm}} \sim e^{c\omega_0/\omega_{\text{local}}}	CTC lifetime
Spectral ratio	R_k = S(\omega_0/k)/S(\omega_0)	Detection criterion

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15. Open Theoretical Questions

1. Universality: Do cognitive time crystals belong to a universality class? What are the critical exponents?

2. Quantum-classical crossover: At what scale does quantum coherence matter for CTC dynamics?

3. Topological protection: Can topological invariants protect CTC phases?

4. Optimal architecture: What network topology maximizes CTC stability?

5. Information capacity: How does CTC phase affect information storage capacity?

6. Multi-stability: Can multiple CTC phases coexist (different k values)?

7. Phase transitions: What is the order of the CTC transition (first-order vs. continuous)?

8. Role of inhibition: How does E-I balance affect CTC formation?

9. Synaptic plasticity: How do learning rules interact with CTC dynamics?

10. Cross-frequency coupling: Can hierarchical CTCs (multiple k simultaneously) exist?

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This mathematical framework provides the foundation for rigorously testing the cognitive time crystal hypothesis. Each equation makes specific, quantitative predictions that can be validated experimentally or computationally.