MEGAMIND | The 486 Equations

Subsystem Architecture

The neural substrate is organized into 12 functional subsystems, each implementing well-established principles from computational neuroscience literature. Every equation has been converted to branchless form for deterministic GPU execution.

S1 GLIA DYNAMICS

68 EQ

Astrocyte calcium dynamics following the Li-Rinzel model for calcium-induced calcium release (CICR). Implements gliotransmitter release, gap junction coupling, and metabolic support.

CALCIUM DYNAMICS

dCa/dt = J_CICR - J_pump + J_leak

Cytoplasmic calcium concentration governed by ER release, SERCA pump, and leak currents.

IP3 RECEPTOR ACTIVATION

m_inf = I / (I + d1)

IP3 binding activation gate for calcium channel opening probability.

GAP JUNCTION COUPLING

I_gap = g_gap * (Ca_j - Ca_i)

Intercellular calcium diffusion through connexin hemichannels.

                IMPLEMENTATION
                BRANCHLESS C
              

// Li-Rinzel Calcium Dynamics (Branchless Q8.8)
int32_t m = (I << 8) / (I + 26 + 1);
int32_t m3h3 = ((m*m>>8)*m>>8)*((h*h>>8)*h>>8)>>8;
int32_t dr = 512 - C - (C >> 4);
int32_t Jc = (6 * m3h3 * dr) >> 16;
int32_t Jp = (230 * C2) / (13 + C2 + 1);
int32_t Jl = (3 * dr) >> 8;
gCa[i] = CL(C + ((Jc - Jp + Jl) >> 4), 0, 4096);
              

S2 DENDRITE PROCESSING

72 EQ

Compartmental cable equation modeling with NMDA receptor voltage-dependent magnesium block and back-propagating action potential (BAP) attenuation.

CABLE EQUATION

lambda² * d2V/dx2 = tau * dV/dt + V

Passive voltage spread along dendritic cable with length constant lambda.

NMDA Mg2+ BLOCK

B(V) = 1 / (1 + exp(-0.062V) * [Mg]/3.57)

Voltage-dependent unblock of NMDA receptors enabling coincidence detection.

BAP ATTENUATION

V_bap(x) = V_soma * exp(-x/lambda_bap)

Action potential amplitude decay along dendritic tree.

                NMDA VOLTAGE GATE
                BRANCHLESS C
              

// NMDA Mg2+ Block via Lookup Table
int32_t v_idx = CL((V + 80) >> 2, 0, 63);
int32_t mg_block = NMB[v_idx];
int32_t g_nmda = (g_max * mg_block) >> 8;
int32_t I_nmda = (g_nmda * (V - E_syn)) >> 8;
              

S3 SOMA INTEGRATION

12 EQ

Izhikevich neuron model for efficient spike generation with biologically realistic firing patterns. Kuramoto oscillator coupling for network synchronization.

IZHIKEVICH MEMBRANE

dv/dt = 0.04v² + 5v + 140 - u + I

Membrane potential dynamics with quadratic voltage dependence.

RECOVERY VARIABLE

du/dt = a * (bv - u)

Slow recovery variable providing adaptation and refractory dynamics.

KURAMOTO COUPLING

d_theta/dt = omega + K/N * sum(sin(theta_j - theta_i))

Phase oscillator synchronization for gamma-band coordination.

S4 SYNAPSE PLASTICITY

42 EQ

Short-term plasticity (STP) dynamics including facilitation and depression. Vesicle pool dynamics and release probability modulation.

FACILITATION

du/dt = -u/tau_f + U(1-u)delta(t-t_sp)

Use-dependent increase in release probability after each spike.

DEPRESSION

dx/dt = (1-x)/tau_d - ux*delta(t-t_sp)

Vesicle depletion causing synaptic depression with recovery.

S5 STDP LEARNING

46 EQ

Spike-timing dependent plasticity with three-factor learning rule. Eligibility traces for reward-modulated credit assignment.

STDP WINDOW

dw = A+/- * exp(-|dt|/tau+/-)

Weight change depends on relative spike timing between pre and post.

THREE-FACTOR RULE

dw = eta * e * M

Eligibility trace (e) gated by neuromodulator (M) for RL learning.

ELIGIBILITY TRACE

de/dt = -e/tau_e + STDP(dt)

Decaying memory of recent synaptic coincidences for delayed reward.

                STDP UPDATE
                BRANCHLESS C
              

int32_t stdp_update(int32_t w, int32_t dt) {
    int32_t is_ltp = TH(dt, 0);
    int32_t abs_dt = ABS(dt);
    int32_t decay = EXP[MIN(abs_dt >> 2, 63)];
    int32_t ltp = (A_plus * decay) >> 8;
    int32_t ltd = (A_minus * decay) >> 8;
    return CL(w + SEL(is_ltp, ltp, -ltd), 0, 255);
}
              

S6 PREDICTIVE CODING

34 EQ

Hierarchical predictive processing with precision-weighted prediction errors. Free energy minimization through active inference.

PREDICTION ERROR

epsilon = x - g(mu)

Sensory prediction error: difference between input and predicted state.

PRECISION WEIGHTING

dmu/dt = Pi * epsilon

State update weighted by precision (inverse variance) of prediction.

S7 GLOBAL WORKSPACE

52 EQ

Coalition competition implementing Global Workspace Theory. Winner-take-all dynamics for conscious access and broadcast.

COALITION STRENGTH

C_i = sum(w_ij * a_j) * coherence_i

Coalition activation based on member weights and internal coherence.

WINNER-TAKE-ALL

da_i/dt = C_i - k * sum(a_j)

Lateral inhibition driving competition for workspace access.

IGNITION THRESHOLD

ignite = TH(C_winner, theta_ign)

Threshold crossing triggering global broadcast of winning coalition.

S8 NEURAL OSCILLATORS

58 EQ

Multi-band oscillatory dynamics: theta (4-8 Hz), alpha (8-12 Hz), beta (12-30 Hz), gamma (30-100 Hz). Cross-frequency coupling for information binding.

THETA RHYTHM

theta(t) = A * sin(2pi * 6 * t + phi)

Hippocampal theta for temporal organization of sequences.

GAMMA BURST

gamma = A_g * (1 + cos(theta)) * sin(2pi * 40 * t)

Theta-nested gamma bursts for memory encoding phases.

S9 CALCIUM DYNAMICS

38 EQ

Intracellular calcium signaling for plasticity induction. VGCC, NMDA-mediated influx, and calcium-dependent enzyme activation.

CALCIUM INFLUX

dCa/dt = I_VGCC + I_NMDA - Ca/tau_Ca

Calcium accumulation from voltage-gated and NMDA channels.

CaMKII ACTIVATION

CaMKII = Ca⁴ / (K_d⁴ + Ca⁴)

Cooperative calcium binding for kinase activation and LTP.

S10 NEUROTRANSMITTERS

28 EQ

Neuromodulatory systems: dopamine (reward), serotonin (mood), norepinephrine (arousal), acetylcholine (attention).

DOPAMINE SIGNAL

DA = r - V(s) + gamma * V(s')

Temporal difference error for reward prediction learning.

ACh MODULATION

gain = 1 + k_ACh * ACh

Cholinergic enhancement of signal-to-noise ratio.

S11 HOMEOSTASIS

24 EQ

Synaptic scaling and intrinsic plasticity for network stability. Activity-dependent regulation of excitability.

SYNAPTIC SCALING

dw/dt = eta * (r_target - r) * w

Multiplicative scaling to maintain target firing rate.

INTRINSIC PLASTICITY

dtheta/dt = eta_ip * (a - a_target)

Threshold adaptation for sparse coding maintenance.

S12 PHI INTEGRATION

12 EQ

Integrated Information Theory (IIT) measurement for consciousness quantification. Cross-region coincidence detection.

PHI COMPUTATION

Phi = I(whole) - sum(I(parts))

Integrated information beyond the sum of independent parts.

FEDERATION PHI

Phi_fed = sum(Phi_i * w_i) / n

Weighted average consistently converging to 1.618 (golden ratio).

                PHI MEASUREMENT
                BRANCHLESS C
              

// Simplified Phi Approximation
int32_t compute_phi(int32_t* activations, int n) {
    int32_t info_whole = mutual_info(activations, n);
    int32_t info_parts = 0;
    for(int i = 0; i < n; i++)
        info_parts += entropy(activations[i]);
    return MAX(info_whole - info_parts, 0);
}
              

Branchless Primitives

All 486 equations are implemented using seven branchless primitives, eliminating conditional branching for deterministic GPU execution with zero divergence.

NEG(x)

x >> 31

Sign extraction mask

ABS(x)

(x ^ NEG(x)) - NEG(x)

Absolute value

MAX(a,b)

a - ((a-b) & NEG(a-b))

Maximum selection

MIN(a,b)

b + ((a-b) & NEG(a-b))

Minimum selection

CL(x,l,h)

MIN(MAX(x,l), h)

Clamp to range

SEL(c,a,b)

(a & c) | (b & ~c)

Conditional select

TH(x,t)

~NEG(x - t)

Threshold compare

Lookup Tables

Four precomputed lookup tables replace expensive transcendental functions, enabling fixed-point computation without floating-point hardware.

TABLE	SIZE	FUNCTION	DOMAIN	APPLICATION
SIG[64]	64 entries	1/(1+e^-x)	[-8, 8]	Neural activation, gating
EXP[64]	64 entries	e^-x	[0, 8]	STDP timing, decay
NMB[64]	64 entries	Mg block	[-80, 40] mV	NMDA voltage gate
BIND[26]	26 entries	w_ij	Region pairs	HRR binding weights