Attractors in Neural Network Circuits: Beauty and Chaos

is one factor in frequent between recollections, oscillating chemical reactions and double pendulums? All these programs have a basin of attraction for attainable states, like a magnet that pulls the system in direction of sure trajectories. Advanced programs with a number of inputs often evolve over time, producing intricate and typically chaotic behaviors. Attractors signify the long-term behavioral sample of dynamical programs — a sample to which a system converges over time no matter its preliminary circumstances. 

Neural networks have turn into ubiquitous in our present Artificial Intelligence period, sometimes serving as highly effective instruments for illustration extraction and sample recognition. Nonetheless, these programs can be considered by one other fascinating lens: as dynamical programs that evolve and converge to a manifold of states over time. When carried out with suggestions loops, even easy neural networks can produce strikingly stunning attractors, starting from restrict cycles to chaotic buildings.

Neural Networks as Dynamical Techniques

Whereas neural networks typically sense are mostly identified for embedding extraction duties, they can be considered as dynamical programs. A dynamical system describes how factors in a state house evolve over time in accordance with a set algorithm or forces. Within the context of neural networks, the state house consists of the activation patterns of neurons, and the evolution rule is decided by the community’s weights, biases, activation features, and different methods.

Conventional NNs are optimized by way of gradient descent to search out its endstate of convergence. Nonetheless, once we introduce suggestions — connecting the output again to the enter — the community turns into a recurrent system with a special sort of temporal dynamic. These dynamics can exhibit a variety of behaviors, from easy convergence to a set level to advanced chaotic patterns.

Understanding Attractors

An attractor is a set of states towards which a system tends to evolve from all kinds of beginning circumstances. As soon as a system reaches an attractor, it stays inside that set of states until perturbed by an exterior pressure. Attractors are certainly deeply concerned in forming recollections [1], oscillating chemical reactions [2], and different nonlinear dynamical programs. 

Sorts of Attractors

Dynamical Systems can exhibit a number of kinds of attractors, every with distinct traits:

• Level Attractors: the only type, the place the system converges to a single fastened level no matter beginning circumstances. This represents a steady equilibrium state.
• Restrict Cycles: the system settles right into a repeating periodic orbit, forming a closed loop in part house. This represents oscillatory habits with a set interval.
• Toroidal (Quasiperiodic) Attractors: the system follows trajectories that wind round a donut-like construction within the part house. In contrast to restrict cycles, these trajectories by no means actually repeat however they continue to be sure to a particular area.
• Unusual (Chaotic) Attractors: characterised by aperiodic habits that by no means repeats precisely but stays bounded inside a finite area of part house. These attractors exhibit delicate dependence on preliminary circumstances, the place a tiny distinction will introduce important penalties over time — an indicator of chaos. Suppose butterfly impact.

Setup

Within the following part, we are going to dive deeper into an instance of a quite simple NN structure able to stated habits, and reveal some fairly examples. We’ll contact on Lyapunov exponents, and supply implementation for individuals who want to experiment with producing their very own Neural Network attractor artwork (and never within the generative AI sense).

We’ll use a grossly simplified one-layer NN with a suggestions loop. The structure consists of:

1. Enter Layer:
   • Array of dimension D (right here 16-32) inputs
   • We’ll unconventionally label them as y₁, y₂, y₃, …, y_D to spotlight that these are mapped from the outputs
   • Acts as a shift register that shops earlier outputs
2. Hidden Layer:
   • Incorporates N neurons (right here fewer than D, ~4-8)
   • We’ll label them x₁, x₂, …, x_N
   • tanh() activation is utilized for squashing
3. Output Layer
   • Single output neuron (y₀)
   • Combines the hidden layer outputs with biases — sometimes, we use biases to offset outputs by including them; right here, we used them for scaling, so they’re factually an array of weights
4. Connections:
   • Enter to Hidden: Weight matrix w[i,j] (randomly initialized between -1 and 1)
   • Hidden to Output: Bias weights b[i] (randomly initialized between 0 and s)
5. Suggestions Loop:
   • The output y₀ is fed again to the enter layer, making a dynamic map
   • Acts as a shift register (y₁ = earlier y₀, y₂ = earlier y₁, and so forth.)
   • This suggestions is what creates the dynamical system habits
6. Key Formulation:
   • Hidden layer: u[i] = Σ(w[i,j] * y[j]); x[i] = tanh(u[i])
   • Output: y₀ = Σ(b[i] * x[i])

The vital features that make this community generate attractors:

• The suggestions loop turns a easy feedforward community right into a dynamical system
• The nonlinear activation operate (tanh) allows advanced behaviors
• The random weight initialization (managed by the random seed) creates totally different attractor patterns
• The scaling issue s impacts the dynamics of the system and may push it into chaotic regimes

In an effort to examine how susceptible the system is to chaos, we are going to calculate the Lyapunov exponents for various units of parameters. Lyapunov exponent is a measure of the instability of a dynamical system…

[delta Z(t)| approx e^{lambda t} |delta (Z(0))|]

[lambda = n_t sum_{k=0}^{n_t-1} ln frac{|Delta y_{k+1}|}]

…the place n_t​ is a lot of time steps, Δy_ok ​is a distance between the states y(x_i) and y(x_i+ϵ) at a time limit; ΔZ(0) represents an preliminary infinitesimal (very small) separation between two close by beginning factors, and ΔZ(t) is the separation after time t. For steady programs converging to a set level or a steady attractor this parameter is lower than 0, for unstable (diverging, and, due to this fact, chaotic programs) it’s better than 0.

Let’s code it up! We’ll solely use NumPy and default Python libraries for the implementation.

import numpy as np
from typing import Tuple, Record, Elective


class NeuralAttractor:
    """
    
    N : int
        Variety of neurons within the hidden layer
    D : int
        Dimension of the enter vector
    s : float
        Scaling issue for the output

    """
    
    def __init__(self, N: int = 4, D: int = 16, s: float = 0.75, seed: Elective[int] = 
None):
        self.N = N
        self.D = D
        self.s = s
        
        if seed is just not None:
            np.random.seed(seed)
        
        # Initialize weights and biases
        self.w = 2.0 * np.random.random((N, D)) - 1.0  # Uniform in [-1, 1]
        self.b = s * np.random.random(N)  # Uniform in [0, s]
        
        # Initialize state vector buildings
        self.x = np.zeros(N)  # Neuron states
        self.y = np.zeros(D)  # Enter vector

We initialize the NeuralAttractor class with some primary parameters — variety of neurons within the hidden layer, variety of components within the enter array, scaling issue for the output, and random seed. We proceed to initialize the weights and biases randomly, and x and y states. These weights and biases is not going to be optimized — they’ll keep put, no gradient descent this time.

    def reset(self, init_value: float = 0.001):
        """Reset the community state to preliminary circumstances."""
        self.x = np.ones(self.N) * init_value
        self.y = np.zeros(self.D)
        
    def iterate(self) -> np.ndarray:
        """
        Carry out one iteration of the community and return the neuron outputs.
        
        """
        # Calculate the output y0
        y0 = np.sum(self.b * self.x)
        
        # Shift the enter vector
        self.y[1:] = self.y[:-1]
        self.y[0] = y0
        
        # Calculate the neuron inputs and apply activation fn
        for i in vary(self.N):
            u = np.sum(self.w[i] * self.y)
            self.x[i] = np.tanh(u)
            
        return self.x.copy()

Subsequent, we are going to outline the iteration logic. We begin each iteration with the suggestions loop — we implement the shift register circuit by shifting all y components to the precise, and compute the newest y₀ output to position it into the primary component of the enter.

    def generate_trajectory(self, tmax: int, discard: int = 0) -> Tuple[np.ndarray, 
np.ndarray]:
        """
        Generate a trajectory of the states for tmax iterations.
        
        -----------
        tmax : int
            Complete variety of iterations
        discard : int
            Variety of preliminary iterations to discard

        """
        self.reset()
        
        # Discard preliminary transient
        for _ in vary(discard):
            self.iterate()
        
        x1_traj = np.zeros(tmax)
        x2_traj = np.zeros(tmax)
        
        for t in vary(tmax):
            x = self.iterate()
            x1_traj[t] = x[0]
            x2_traj[t] = x[1]
            
        return x1_traj, x2_traj

Now, we outline the operate that can iterate our community map over the tmax variety of time steps and output the states of the primary two hidden neurons for visualization. We are able to use any hidden neurons, and we might even visualize 3D state house, however we are going to restrict our creativeness to 2 dimensions.

That is the gist of the system. Now, we are going to simply outline some line and phase magic for fairly visualizations.

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.collections as mcoll
import matplotlib.path as mpath
from typing import Tuple, Elective, Callable


def make_segments(x: np.ndarray, y: np.ndarray) -> np.ndarray:
    """
    Create record of line segments from x and y coordinates.
    
    -----------
    x : np.ndarray
        X coordinates
    y : np.ndarray
        Y coordinates

    """
    factors = np.array([x, y]).T.reshape(-1, 1, 2)
    segments = np.concatenate([points[:-1], factors[1:]], axis=1)
    return segments


def colorline(
    x: np.ndarray,
    y: np.ndarray,
    z: Elective[np.ndarray] = None,
    cmap = plt.get_cmap("jet"),
    norm = plt.Normalize(0.0, 1.0),
    linewidth: float = 1.0,
    alpha: float = 0.05,
    ax = None
):
    """
    Plot a coloured line with coordinates x and y.
    
    -----------
    x : np.ndarray
        X coordinates
    y : np.ndarray
        Y coordinates

    """
    if ax is None:
        ax = plt.gca()
        
    if z is None:
        z = np.linspace(0.0, 1.0, len(x))
    
    segments = make_segments(x, y)
    lc = mcoll.LineCollection(
        segments, array=z, cmap=cmap, norm=norm, linewidth=linewidth, alpha=alpha
    )
    ax.add_collection(lc)
    
    return lc


def plot_attractor_trajectory(
    x: np.ndarray,
    y: np.ndarray,
    skip_value: int = 16,
    color_function: Elective[Callable] = None,
    cmap = plt.get_cmap("Spectral"),
    linewidth: float = 0.1,
    alpha: float = 0.1,
    figsize: Tuple[float, float] = (10, 10),
    interpolate_steps: int = 3,
    output_path: Elective[str] = None,
    dpi: int = 300,
    present: bool = True
):
    """
    Plot an attractor trajectory.
    
    Parameters:
    -----------
    x : np.ndarray
        X coordinates
    y : np.ndarray
        Y coordinates
    skip_value : int
        Variety of factors to skip for sparser plotting

    """
    fig, ax = plt.subplots(figsize=figsize)
    
    if interpolate_steps > 1:
        path = mpath.Path(np.column_stack([x, y]))
        verts = path.interpolated(steps=interpolate_steps).vertices
        x, y = verts[:, 0], verts[:, 1]
    
    x_plot = x[::skip_value]
    y_plot = y[::skip_value]
    
    if color_function is None:
        z = abs(np.sin(1.6 * y_plot + 0.4 * x_plot))
    else:
        z = color_function(x_plot, y_plot)
    
    colorline(x_plot, y_plot, z, cmap=cmap, linewidth=linewidth, alpha=alpha, ax=ax)
    
    ax.set_xlim(x.min(), x.max())
    ax.set_ylim(y.min(), y.max())
    
    ax.set_axis_off()
    ax.set_aspect('equal')
    
    plt.tight_layout()
    
    if output_path:
        fig.savefig(output_path, dpi=dpi, bbox_inches='tight')

    return fig

The features written above will take the generated state house trajectories and visualize them. As a result of the state house could also be densely stuffed, we are going to skip each eighth, sixteenth or 32th time level to sparsify our vectors. We additionally don’t wish to plot these in a single strong shade, due to this fact we’re coding the colour as a periodic operate (np.sin(1.6 * y_plot + 0.4 * x_plot)) primarily based on the x and y coordinates of the determine axis. The multipliers for the coordinates are arbitrary and occur to generate good clean shade maps, to your liking.

N = 4
D = 32
s = 0.22
seed=174658140

tmax = 100000
discard = 1000

nn = NeuralAttractor(N, D, s, seed=seed)

# Generate trajectory
x1, x2 = nn.generate_trajectory(tmax, discard)

plot_attractor_trajectory(
    x1, x2,
    output_path='trajectory.png',
)

After defining the NN and iteration parameters, we will generate the state house trajectories. If we spend sufficient time poking round with parameters, we are going to discover one thing cool (I promise!). If guide parameter grid search labor is just not precisely our factor, we might add a operate that checks what proportion of the state house is roofed over time. If after t = 100,000 iterations (besides the preliminary 1,000 “heat up” time steps) we solely touched a slim vary of values of the state house, we’re possible caught in a degree. As soon as we discovered an attractor that isn’t so shy to take up extra state house, we will plot it utilizing default plotting params:

One of many steady kinds of attractors is the restrict cycle attractor (parameters: N = 4, D = 32, s = 0.22, seed = 174658140). It appears to be like like a single, closed loop trajectory in part house. The orbit follows a daily, periodic path over time sequence. I cannot embody the code for Lyapunov exponent calculation right here to deal with the visible facet of the generated attractors extra, however one can discover it below this link, if . The Lyapunov exponent for this attractor (λ=−3.65) is damaging, indicating stability: mathematically, this exponent will result in the state of the system decaying, or converging, to this basin of attraction over time.

If we preserve rising the scaling issue, we usually tend to tune up the values within the circuit, and maybe extra prone to discover one thing attention-grabbing.

Right here is the toroidal (quasiperiodic) attractor (parameters: N = 4, D = 32, s = 0.55, seed = 3160697950). It nonetheless has an ordered construction of sheets that wrap round in organized, quasiperiodic patterns. The Lyapunov exponent for this attractor has a better worth, however continues to be damaging (λ=−0.20).

As we additional enhance the scaling issue s, the system turns into extra liable to chaos. The unusual (chaotic) attractor emerges with the next parameters: N = 4, D = 16, s = 1.4, seed = 174658140). It’s characterised by an erratic, unpredictable sample of trajectories that by no means repeat. The Lyapunov exponent for this attractor is constructive (λ=0.32), indicating instability (divergence from an initially very shut state over time) and chaotic habits. That is the “butterfly impact” attractor.

As we additional enhance the scaling issue s, the system turns into extra liable to chaos. The unusual (chaotic) attractor emerges with the next parameters: N = 4, D = 16, s = 1.4, seed = 174658140. It’s characterised by an erratic, unpredictable sample of trajectories that by no means repeat. The Lyapunov exponent for this attractor is constructive (λ=0.32), indicating instability (divergence from an initially very shut state over time) and chaotic habits. That is the “butterfly impact” attractor.

Simply one other affirmation that aesthetics could be very mathematical, and vice versa. Essentially the most visually compelling attractors usually exist on the fringe of chaos — give it some thought for a second! These buildings are advanced sufficient to exhibit intricate habits, but ordered sufficient to keep up coherence. This resonates with observations from varied artwork types, the place steadiness between order and unpredictability usually creates probably the most participating experiences.

An interactive widget to generate and visualize these attractors is offered here. The supply code is available, too, and invitations additional exploration. The concepts behind this mission have been largely impressed by the work of J.C. Sprott [3]. 

References

[1] B. Poucet and E. Save, Attractors in Reminiscence (2005), Science DOI:10.1126/science.1112555.

[2] Y.J.F. Kpomahou et al., Chaotic Behaviors and Coexisting Attractors in a New Nonlinear Dissipative Parametric Chemical Oscillator (2022), Complexity DOI:10.1155/2022/9350516.

[3] J.C. Sprott, Synthetic Neural Web Attractors (1998), Computer systems & Graphics DOI:10.1016/S0097-8493(97)00089-7.