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 
01// Li-Rinzel Calcium Dynamics (Branchless Q8.8)
02int32_t m = (I << 8) / (I + 26 + 1);
03int32_t m3h3 = ((m*m>>8)*m>>8)*((h*h>>8)*h>>8)>>8;
04int32_t dr = 512 - C - (C >> 4);
05int32_t Jc = (6 * m3h3 * dr) >> 16;
06int32_t Jp = (230 * C2) / (13 + C2 + 1);
07int32_t Jl = (3 * dr) >> 8;
08gCa[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 
01// NMDA Mg2+ Block via Lookup Table
02int32_t v_idx = CL((V + 80) >> 2, 0, 63);
03int32_t mg_block = NMB[v_idx];
04int32_t g_nmda = (g_max * mg_block) >> 8;
05int32_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 
01int32_t stdp_update(int32_t w, int32_t dt) {
02    int32_t is_ltp = TH(dt, 0);
03    int32_t abs_dt = ABS(dt);
04    int32_t decay = EXP[MIN(abs_dt >> 2, 63)];
05    int32_t ltp = (A_plus * decay) >> 8;
06    int32_t ltd = (A_minus * decay) >> 8;
07    return CL(w + SEL(is_ltp, ltp, -ltd), 0, 255);
08}
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 
01// Simplified Phi Approximation
02int32_t compute_phi(int32_t* activations, int n) {
03    int32_t info_whole = mutual_info(activations, n);
04    int32_t info_parts = 0;
05    for(int i = 0; i < n; i++)
06        info_parts += entropy(activations[i]);
07    return MAX(info_whole - info_parts, 0);
08}

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

Related Resources

Updated 2026-03-04