2. Mathematical Formulations & Neurogenetic Architecture
The cognitive apparatus of each organism maps sensory afferents through intermediate interneurons to motor efferents. The complete state space consists of $N_s = 7$ sensory inputs, $N_h$ internal hidden neurons (with user-configurable cardinality $0 \le N_h \le 6$), and $N_a = 5$ motor channels.
Sensory Vector Formulation
At discrete time-step $t \in [0, T_{\text{step}}]$, creature $i$ computes a normalized sensory vector $\mathbf{s}_i(t) \in \mathbb{R}^7$:
$$\mathbf{s}_i(t) = \begin{bmatrix}
x_i / W \\
y_i / H \\
t / T_{\text{step}} \\
\min(x_i, y_i, W - x_i, H - y_i) / \max(W, H) \\
\sin(2\pi t / \tau_{\text{osc}}) \\
\rho_{\text{local}}(x_i, y_i) \\
\Phi_{\text{goal}}(x_i, y_i)
\end{bmatrix}$$
Here, $W$ and $H$ delineate virtual world bounds, $\tau_{\text{osc}} = 50$ ticks defines the circadian pacing period, $\rho_{\text{local}} \in [0, 1]$ samples local crowd density within neighborhood radius $R_c = 40\,\text{px}$, and $\Phi_{\text{goal}}$ captures the proximity field strength toward the scenario target.
Synaptic Activation & Hyperbolic Tangent Transfer
Every gene on the chromosome encodes a directional synaptic link consisting of a source classification (Sensor or Hidden), source identifier, sink classification (Hidden or Action), sink identifier, and a continuous scalar synaptic weight $w \in [-2.0, 2.0]$. Internal interneurons update according to:
$$h_j(t) = \tanh \left( \sum_{k \in \mathcal{S} \to h_j} w_{kj} s_k(t) + \sum_{m \in \mathcal{H} \to h_j} w_{mj} h_m(t - 1) \right)$$
Outputs $\mathbf{a}_i(t) \in [-1, 1]^5$ are subsequently computed via feedforward synaptic integration:
$$a_l(t) = \tanh \left( \sum_{k \in \mathcal{S} \to a_l} w_{kl} s_k(t) + \sum_{j \in \mathcal{H} \to a_l} w_{jl} h_j(t) \right)$$
Kinematic Integration & Viscous Damping
To model biological inertia, creature velocity $\mathbf{v}(t) = (v_x, v_y)$ undergoes exponential smoothing with fluid damping coefficient $\alpha = 0.85$:
$$\mathbf{v}(t) = \alpha \mathbf{v}(t - 1) + (1 - \alpha) \cdot v_{\text{max}} \begin{bmatrix}
a_0(t) + a_2(t) \cos \theta + a_4(t) \xi_x \\
a_1(t) + a_2(t) \sin \theta + a_4(t) \xi_y
\end{bmatrix}$$
Where $\theta = \operatorname{atan2}(v_y, v_x)$ denotes the instantaneous heading angle, $a_2$ governs longitudinal propulsion surge, $a_4$ controls stochastic Brownian foraging excursion driven by uniform noise $\xi \sim \mathcal{U}(-0.5, 0.5)$, and $v_{\text{max}}$ is the user-regulated speed parameter.
Selection Mechanics & Genetic Crossover
At generational boundary $t = T_{\text{step}}$, natural selection applies a discrete spatial truncation filter $\mathbb{I}_{\text{niche}}(\mathbf{r}_i) \in \{0, 1\}$. Surviving specimens $\mathcal{P}_{\text{surv}}$ enter the reproduction pool:
$$g_{\text{child}}[k] = \begin{cases}
g_{\text{parentA}}[k], & \text{if } k < k_{\text{cross}} \\
g_{\text{parentB}}[k], & \text{if } k \ge k_{\text{cross}}
\end{cases}$$
Point mutation operator $\mathcal{M}$ perturbs individual genes with probability $P_{\mu}$. With probability $0.6$, the synaptic weight experiences Gaussian jitter $w \leftarrow \operatorname{clamp}(w + \mathcal{N}(0, 0.35^2), -2, 2)$, while with probability $0.4$, structural reconvolution reassigns source or sink node addresses, inducing topological neuroplasticity across macro-evolutionary timescales.
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Dr. Yuri Beno.