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@brief Compute weight gradient: ∂L/∂w_ij = δ_j · x_i

@brief Three-factor learning: Δw = η·x_pre·x_post·R

@brief Eligibility trace update: e(t+1) = λ·e(t) + x_pre·x_post

@brief TD error: δ = r + γV(s') - V(s)

@brief Policy gradient: ∇J ≈ Σ(∇log π(a|s)·R)

dΦ/dt = α(Φ_target - Φ) + β∇²Φ + γξ(t)

Where: α=decay, β=diffusion, γ=noise, ξ=Wiener process

ds² = gᵢⱼ(θ)dθⁱdθʲ  (Fisher information metric)

gᵢⱼ = E[∂log p/∂θⁱ · ∂log p/∂θʲ]

F = E_q[log q(z) - log p(x,z)]

= D_KL(q||p) - log p(x)

ε = x - μ  (prediction error)

μ_new = μ + α·ε  (update prediction)

ds² = gᵢⱼdθⁱdθʲ

Natural gradient: ∇̃L = G⁻¹∇L

F = D_KL(q||p) - log p(x|z)

KL = 0.5 * (log(σ_p²/σ_q²) + σ_q²/σ_p² + (μ_q - μ_p)²/σ_p² - 1)

∂KL/∂μ = (μ - μ_prior) / σ_prior²

Hebbian update: ΔW = η·x·y

BCM rule: ΔW = η·y·(y - θ)·x where θ is sliding threshold

L = -log(exp(sim(z_i, z_j)/τ) / Σ_k exp(sim(z_i, z_k)/τ))

ds² = g_ij dx^i dx^j

g_ij(θ) = E[∂log p(x|θ)/∂θ_i · ∂log p(x|θ)/∂θ_j]

d²x^μ/dt² + Γ^μ_νρ (dx^ν/dt)(dx^ρ/dt) = 0

∂g_ij/∂t = -2R_ij

H(X) = -Σ p(x) log p(x)

I(X;Y) = H(X) + H(Y) - H(X,Y)

D_KL(P||Q) = Σ P(x) log(P(x)/Q(x))

Φ = min_{partition} I(past; future) - Σ I(past_i; future_i)

dX_t = μ(X_t, t)dt + σ(X_t, t)dW_t

∂p/∂t = -∂(μp)/∂x + ½∂²(σ²p)/∂x²

dX_t = θ(μ - X_t)dt + σdW_t

W_p(μ,ν) = (inf_{γ∈Γ(μ,ν)} ∫|x-y|^p dγ)^(1/p)

W_1(μ,ν) = sup_{Lip(f)≤1} ∫f d(μ-ν)

g_ij = E[∂log p/∂θ_i · ∂log p/∂θ_j]

Γ_{ijk} = ½(∂g_{ij}/∂x^k + ∂g_{ik}/∂x^j - ∂g_{jk}/∂x^i)

a^μ = -Γ^μ_νρ v^ν v^ρ

R = 2K where K is Gaussian curvature

exp_p(v) ≈ p + v - ½Γ(v,v)

@brief Shannon entropy H(X) = -Σ p(x) log p(x)

@brief Binary entropy H(p) = -p log p - (1-p) log(1-p)

@brief Cross entropy H(P,Q) = -Σ P(x) log Q(x)

@brief KL Divergence D_KL(P||Q) = Σ P(x) log(P(x)/Q(x))

@brief Mutual information I(X;Y) = H(X) + H(Y) - H(X,Y)

Φ = I(system) - Σ I(parts)

@brief Itô SDE step: dX = μdt + σdW

@brief Ornstein-Uhlenbeck step: dX = θ(μ - X)dt + σdW

@brief Langevin dynamics step: dX = -∇V(X)dt + √(2T)dW

W_1(μ,ν) = ∫|F_μ^{-1}(t) - F_ν^{-1}(t)| dt

@brief Consciousness gradient: ∂Φ/∂x = (x-1)/((1+|x-1|)²) · sign(x-1)

@brief Lyapunov function gradient: ∂V/∂Ω = 2(Ω - Ω*)

@brief Free energy gradient: ∂F/∂x = 2(x - μ) - T·∂S/∂x

@brief Natural gradient (Fisher-Rao): ∇̃f = G^{-1}∇f

@brief Sigmoid with derivative: σ(x), σ'(x) = σ(x)(1-σ(x))

@brief Tanh with derivative: tanh(x), tanh'(x) = 1 - tanh²(x)

@brief Outer product: M = α·u⊗v + M

LEVEL 1: NEURON (Single Mind = Single Neuron)

LEVEL 2: NEURAL NETWORK (Mind Farm = Neural Tissue)

Nernst Equation: E_ion = (RT/zF) * ln([ion]_out/[ion]_in)

V_m = (RT/F) * ln((P_Na[Na]_o + P_K[K]_o + P_Cl[Cl]_i) / (P_Na[Na]_i + P_K[K]_i + P_Cl[Cl]_o))

C_m * dV/dt = I_ext - g_Na*m³*h*(V-E_Na) - g_K*n⁴*(V-E_K) - g_L*(V-E_L)

α_m(V) = 0.1(V+40)/(1-exp(-(V+40)/10))

β_m(V) = 4*exp(-(V+65)/18)

α_h(V) = 0.07*exp(-(V+65)/20)

β_h(V) = 1/(1+exp(-(V+35)/10))

α_n(V) = 0.01(V+55)/(1-exp(-(V+55)/10))

β_n(V) = 0.125*exp(-(V+65)/80)

I_AMPA = g_AMPA * s * (V - E_AMPA)

ds/dt = -s/τ_AMPA + Σ_j δ(t - t_j)

I_NMDA = g_NMDA * s * B(V) * (V - E_NMDA)