Neuromorphic Parametric Oscillators

---
name: neuromorphic-parametric-oscillators
description: "Neuromorphic computing using parametrically-driven oscillators and optical frequency combs. Leverages nonlinear parametric resonance in coupled oscillator networks for energy-efficient neural network inference. Keywords: neuromorphic computing, parametric oscillators, frequency combs, optical computing, neural network inference."
---

# Neuromorphic Computing with Parametrically-Driven Oscillators

Research methodology from arXiv:2604.21861v1 using parametrically-driven oscillators and optical frequency combs for ultra-fast, energy-efficient neural network inference.

## Overview

This skill implements **neuromorphic computing based on parametrically-driven oscillators**, leveraging **nonlinear parametric resonance** in coupled oscillator networks to perform neural network inference with:

- **Orders of magnitude lower energy** than digital implementations
- **Ultra-fast processing** through optical frequency combs
- **High-dimensional computing** via phase and amplitude encoding
- **Natural fan-in/fan-out** connectivity

## Core Technology

### Parametric Resonance

**Parametric resonance** occurs when driving frequency = 2 × Natural frequency:

```
Natural frequency: f₀
Driving frequency: f_drive = 2f₀

Parametric gain: d²x/dt² + ω₀²[1 + h cos(2ω₀t)]x = 0

Where:
- h: Modulation depth
- ω₀ = 2πf₀: Natural angular frequency
```

### Optical Frequency Combs

**Frequency comb** provides:
- Multiple equidistant frequency lines
- High-dimensional information encoding
- Parallel processing capability
- Phase-coherent operation

## System Architecture

### Parametric Oscillator Unit

```python
import numpy as np
from scipy.integrate import solve_ivp

class ParametricOscillator:
    """
    Parametric oscillator unit for neuromorphic computing.
    
    Implements: d²x/dt² + γ dx/dt + ω₀²[1 + h cos(2ω₀t)]x + αx³ = F(t)
    """
    
    def __init__(self, params):
        self.omega_0 = params.get('omega_0', 2 * np.pi * 1e9)  # 1 GHz
        self.gamma = params.get('gamma', 0.01 * self.omega_0)  # Damping
        self.h = params.get('h', 0.1)  # Parametric modulation depth
        self.alpha = params.get('alpha', 0.01)  # Nonlinearity
        self.omega_d = 2 * self.omega_0  # Drive frequency
        
    def dynamics(self, t, state, drive_amplitude=0.0, external_force=0.0):
        """
        Oscillator dynamics with parametric drive.
        
        Args:
            t: Time
            state: [x, v] position and velocity
            drive_amplitude: Parametric drive amplitude
            external_force: External input force
        
        Returns:
            derivatives: [dx/dt, dv/dt]
        """
        x, v = state
        
        # Parametric modulation
        modulation = self.h * np.cos(self.omega_d * t)
        
        # Damped parametric oscillator with nonlinearity
        dxdt = v
        dvdt = (-self.gamma * v 
                - self.omega_0**2 * (1 + drive_amplitude * modulation) * x 
                - self.alpha * x**3 
                + external_force)
        
        return [dxdt, dvdt]
    
    def simulate(self, t_span, initial_state, input_signal=None, dt=1e-12):
        """
        Simulate oscillator response.
        
        Args:
            t_span: (t_start, t_end)
            initial_state: [x₀, v₀]
            input_signal: Callable input force function
            dt: Time step
        
        Returns:
            t: Time points
            x: Oscillator state [position, velocity]
        """
        times = np.arange(t_span[0], t_span[1], dt)
        
        def dynamics_with_input(t, state):
            force = input_signal(t) if input_signal else 0.0
            return self.dynamics(t, state, external_force=force)
        
        solution = solve_ivp(
            dynamics_with_input,
            t_span,
            initial_state,
            t_eval=times,
            method='RK45'
        )
        
        return solution.t, solution.y
    
    def compute_phase(self, t, x):
        """
        Compute instantaneous phase using Hilbert transform.
        
        Args:
            t: Time array
            x: Oscillator position
        
        Returns:
            phase: Instantaneous phase
        """
        from scipy.signal import hilbert
        
        analytic_signal = hilbert(x)
        instantaneous_phase = np.unwrap(np.angle(analytic_signal))
        
        return instantaneous_phase
    
    def compute_amplitude(self, t, x):
        """
        Compute instantaneous amplitude.
        
        Args:
            t: Time array
            x: Oscillator position
        
        Returns:
            amplitude: Instantaneous amplitude
        """
        from scipy.signal import hilbert
        
        analytic_signal = hilbert(x)
        amplitude = np.abs(analytic_signal)
        
        return amplitude


class FrequencyCombOscillator:
    """
    Parametric oscillator with frequency comb encoding.
    
    Encodes information in phase and amplitude of comb lines.
    """
    
    def __init__(self, n_comb_lines, f_center=1e9, f_spacing=10e6):
        self.n_comb_lines = n_comb_lines
        self.f_center = f_center
        self.f_spacing = f_spacing
        
        # Comb line frequencies
        self.frequencies = np.array([
            f_center + (i - n_comb_lines//2) * f_spacing
            for i in range(n_comb_lines)
        ])
        
        # Create oscillator for each comb line
        self.oscillators = []
        for f in self.frequencies:
            osc = ParametricOscillator({
                'omega_0': 2 * np.pi * f,
                'gamma': 0.01 * 2 * np.pi * f,
                'h': 0.1
            })
            self.oscillators.append(osc)
        
        self.phase_state = np.zeros(n_comb_lines)
        self.amplitude_state = np.zeros(n_comb_lines)
    
    def encode_input(self, input_vector):
        """
        Encode input vector as phase/amplitude modulation.
        
        Args:
            input_vector: Input features (n_comb_lines,)
        
        Returns:
            phase_mod: Phase modulation for each comb line
            amplitude_mod: Amplitude modulation for each comb line
        """
        # Normalize input
        input_norm = input_vector / (np.max(np.abs(input_vector)) + 1e-8)
        
        # Phase encoding: φ = π * input
        phase_mod = np.pi * input_norm
        
        # Amplitude encoding: A = |input|
        amplitude_mod = np.abs(input_norm)
        
        return phase_mod, amplitude_mod
    
    def compute_comb_response(self, phase_mod, amplitude_mod, duration=1e-6):
        """
        Compute frequency comb response to modulated input.
        
        Args:
            phase_mod: Phase modulation per comb line
            amplitude_mod: Amplitude modulation per comb line
            duration: Simulation duration
        
        Returns:
            output_state: [phases, amplitudes] for each comb line
        """
        phases = np.zeros(self.n_comb_lines)
        amplitudes = np.zeros(self.n_comb_lines)
        
        for i, osc in enumerate(self.oscillators):
            # Apply phase modulation as initial condition
            x0 = amplitude_mod[i] * np.cos(phase_mod[i])
            v0 = -amplitude_mod[i] * osc.omega_0 * np.sin(phase_mod[i])
            
            t, x = osc.simulate(
                (0, duration),
                [x0, v0],
                dt=1e-12
            )
            
            # Extract steady-state phase and amplitude
            phases[i] = osc.compute_phase(t, x[0])[-1]
            amplitudes[i] = osc.compute_amplitude(t, x[0])[-1]
        
        return {'phases': phases, 'amplitudes': amplitudes}
```

### Coupled Oscillator Network

```python
class CoupledParametricNetwork:
    """
    Network of coupled parametric oscillators for neural computation.
    """
    
    def __init__(self, n_neurons, coupling_strength=0.1):
        self.n_neurons = n_neurons
        self.coupling_strength = coupling_strength
        
        # Create oscillators with distributed natural frequencies
        frequencies = np.linspace(0.9e9, 1.1e9, n_neurons)
        self.oscillators = [
            ParametricOscillator({'omega_0': 2 * np.pi * f})
            for f in frequencies
        ]
        
        # Random sparse coupling
        self.coupling = np.random.randn(n_neurons, n_neurons) * coupling_strength
        self.coupling[np.abs(self.coupling) < 0.5] = 0
        np.fill_diagonal(self.coupling, 0)
    
    def network_dynamics(self, t, states, input_currents):
        """
        Coupled network dynamics.
        
        State vector: [x₀, v₀, x₁, v₁, ..., xₙ, vₙ]
        
        Args:
            t: Time
            states: Flattened state vector
            input_currents: Input currents to each neuron
        
        Returns:
            derivatives: State derivatives
        """
        n = self.n_neurons
        derivatives = np.zeros(2 * n)
        
        for i in range(n):
            x_i = states[2*i]
            v_i = states[2*i + 1]
            
            # Single oscillator dynamics
            osc = self.oscillators[i]
            dxdt, dvdt = osc.dynamics(t, [x_i, v_i], external_force=input_currents[i])
            
            # Coupling term
            coupling_force = 0
            for j in range(n):
                if self.coupling[i, j] != 0:
                    x_j = states[2*j]
                    # Phase-dependent coupling
                    phase_diff = np.angle(np.exp(1j * np.arctan2(v_i, x_i)) * 
                                         np.conj(np.exp(1j * np.arctan2(states[2*j+1], x_j))))
                    coupling_force += self.coupling[i, j] * np.sin(phase_diff)
            
            derivatives[2*i] = dxdt
            derivatives[2*i + 1] = dvdt + coupling_force
        
        return derivatives
    
    def simulate_network(self, duration, initial_state=None, input_pattern=None):
        """
        Simulate coupled network dynamics.
        
        Args:
            duration: Simulation duration (s)
            initial_state: Initial network state
            input_pattern: Input pattern to encode
        
        Returns:
            t: Time points
            states: Network states over time
        """
        if initial_state is None:
            initial_state = np.random.randn(2 * self.n_neurons) * 0.1
        
        # Create input currents from pattern
        if input_pattern is not None:
            input_currents = input_pattern * 1e-3  # Scale to appropriate range
        else:
            input_currents = np.zeros(self.n_neurons)
        
        def dynamics_fn(t, y):
            return self.network_dynamics(t, y, input_currents)
        
        solution = solve_ivp(
            dynamics_fn,
            (0, duration),
            initial_state,
            method='RK45',
            dense_output=True
        )
        
        return solution.t, solution.y
```

## Neural Network Implementation

### Phase/Amplitude Neural Network

```python
class OscillatorNeuralNetwork:
    """
    Neural network using parametric oscillator encoding.
    
    Information encoded in phase and amplitude of oscillators.
    """
    
    def __init__(self, layer_sizes, use_frequency_combs=False):
        """
        Args:
            layer_sizes: List of layer dimensions
            use_frequency_combs: Whether to use frequency comb encoding
        """
        self.layer_sizes = layer_sizes
        self.use_frequency_combs = use_frequency_combs
        self.n_layers = len(layer_sizes) - 1
        
        # Initialize weight matrices
        self.weights = []
        for i in range(self.n_layers):
            # Complex weights for phase/amplitude
            W = (np.random.randn(layer_sizes[i+1], layer_sizes[i]) + 
                 1j * np.random.randn(layer_sizes[i+1], layer_sizes[i]))
            W = W / np.sqrt(layer_sizes[i])  # Xavier-like initialization
            self.weights.append(W)
        
        # Bias terms
        self.biases = [
            np.random.randn(size) * 0.01
            for size in layer_sizes[1:]
        ]
    
    def encode_input(self, x):
        """
        Encode real-valued input as complex oscillator state.
        
        Args:
            x: Input vector
        
        Returns:
            complex_state: Complex encoding (amplitude * exp(i * phase))
        """
        if self.use_frequency_combs:
            # Use frequency comb oscillator
            n_comb = self.layer_sizes[0]
            comb = FrequencyCombOscillator(n_comb)
            phase, amp = comb.encode_input(x)
            return amp * np.exp(1j * phase)
        else:
            # Simple encoding: amplitude = |x|, phase = angle
            amplitude = np.abs(x)
            phase = np.angle(x + 1e-8) if np.any(np.iscomplex(x)) else np.zeros_like(x)
            return amplitude * np.exp(1j * phase)
    
    def complex_activation(self, z):
        """
        Complex activation function.
        
        Args:
            z: Complex input
        
        Returns:
            activated: Activated complex output
        """
        # ModReLU: z if |z| + b > 0, else 0
        b = 0.1  # Bias
        magnitude = np.abs(z)
        activated = z * np.maximum(magnitude + b, 0) / (magnitude + 1e-8)
        return activated
    
    def forward(self, x):
        """
        Forward pass through the network.
        
        Args:
            x: Input vector
        
        Returns:
            output: Network output
        """
        # Encode input
        h = self.encode_input(x)
        
        # Pass through layers
        for W, b in zip(self.weights[:-1], self.biases[:-1]):
            # Complex linear transformation
            z = W @ h + b
            # Complex activation
            h = self.complex_activation(z)
        
        # Output layer
        z = self.weights[-1] @ h + self.biases[-1]
        
        # Convert to real output (take real part or magnitude)
        output = np.real(z)  # Or np.abs(z) for magnitude
        
        return output
    
    def train(self, X_train, y_train, learning_rate=0.01, epochs=100):
        """
        Train using Wirtinger derivatives (complex backprop).
        
        Args:
            X_train: Training inputs
            y_train: Training targets
            learning_rate: Learning rate
            epochs: Training epochs
        """
        for epoch in range(epochs):
            total_loss = 0
            
            for x, y in zip(X_train, y_train):
                # Forward pass
                activations = [self.encode_input(x)]
                
                for W, b in zip(self.weights, self.biases):
                    z = W @ activations[-1] + b
                    if len(activations) < len(self.weights):
                        a = self.complex_activation(z)
                    else:
                        a = z
                    activations.append(a)
                
                # Compute loss
                output = np.real(activations[-1])
                loss = np.mean((output - y)**2)
                total_loss += loss
                
                # Backprop (simplified - real-valued weights would be needed for full training)
                error = output - y
                
                # Gradient descent on output layer (simplified)
                delta = error
                dW = np.outer(delta, np.conj(activations[-2]))
                self.weights[-1] -= learning_rate * dW
                self.biases[-1] -= learning_rate * delta
            
            if epoch % 10 == 0:
                print(f"Epoch {epoch}, Loss: {total_loss / len(X_train):.4f}")
```

### Reservoir Computing with Parametric Oscillators

```python
class ParametricOscillatorReservoir:
    """
    Reservoir computer using coupled parametric oscillators.
    """
    
    def __init__(self, n_reservoir, n_comb_lines=10, spectral_radius=0.9):
        self.n_reservoir = n_reservoir
        self.n_comb_lines = n_comb_lines
        
        # Frequency comb for high-dimensional encoding
        self.comb = FrequencyCombOscillator(n_comb_lines)
        
        # Coupled oscillator network
        self.network = CoupledParametricNetwork(n_reservoir)
        
        # Scale coupling to spectral radius
        eigenvalues = np.linalg.eigvals(self.network.coupling)
        max_eigenval = np.max(np.abs(eigenvalues))
        self.network.coupling *= spectral_radius / max_eigenval
        
        # Input weights
        self.W_in = np.random.randn(n_reservoir, n_comb_lines) * 0.1
        
        # Readout weights (trained)
        self.W_out = None
    
    def encode_input_sequence(self, sequence):
        """
        Encode input sequence using frequency comb.
        
        Args:
            sequence: Time series input
        
        Returns:
            encoded: Encoded sequence
        """
        encoded = []
        for x in sequence:
            phase, amp = self.comb.encode_input(x)
            response = self.comb.compute_comb_response(phase, amp, duration=1e-9)
            
            # Combine phase and amplitude
            state = np.concatenate([response['phases'], response['amplitudes']])
            encoded.append(state)
        
        return np.array(encoded)
    
    def run_reservoir(self, encoded_sequence, duration_per_step=1e-6):
        """
        Run reservoir dynamics on encoded input.
        
        Args:
            encoded_sequence: Encoded input sequence
            duration_per_step: Duration per time step
        
        Returns:
            states: Reservoir states at each timestep
        """
        states = []
        current_state = np.random.randn(2 * self.n_reservoir) * 0.01
        
        for encoded_input in encoded_sequence:
            # Map encoded input to reservoir currents
            input_currents = self.W_in @ encoded_input[:self.n_comb_lines]
            
            # Run network
            t, new_state = self.network.simulate_network(
                duration_per_step,
                current_state,
                input_currents
            )
            
            # Extract state (use position components)
            reservoir_state = new_state[::2, -1]  # Every other element (positions)
            states.append(reservoir_state)
            
            current_state = new_state[:, -1]
        
        return np.array(states)
    
    def train_readout(self, states, targets, alpha=1.0):
        """
        Train linear readout using ridge regression.
        
        Args:
            states: Reservoir states (time_steps, n_reservoir)
            targets: Target outputs (time_steps, n_outputs)
            alpha: Ridge regularization
        """
        # Ridge regression
        I = np.eye(states.shape[1])
        self.W_out = np.linalg.solve(
            states.T @ states + alpha * I,
            states.T @ targets
        )
    
    def predict(self, states):
        """
        Make predictions using trained readout.
        
        Args:
            states: Reservoir states
        
        Returns:
            predictions: Output predictions
        """
        return states @ self.W_out
```

## Applications

### 1. Time Series Prediction

```python
def predict_mackey_glass(reservoir, data, prediction_steps):
    """
    Predict Mackey-Glass time series.
    
    Args:
        reservoir: Trained ParametricOscillatorReservoir
        data: Training data
        prediction_steps: Steps to predict
    
    Returns:
        predictions: Predicted values
    """
    from scipy.integrate import odeint
    
    # Generate Mackey-Glass data
    def mackey_glass(t, x, beta=0.2, gamma=0.1, tau=17, n=10):
        dxdt = beta * x(t - tau) / (1 + x(t - tau)**n) - gamma * x(t)
        return dxdt
    
    # Encode and run reservoir
    encoded = reservoir.encode_input_sequence(data)
    states = reservoir.run_reservoir(encoded)
    
    # Train readout on last known states
    train_states = states[:-prediction_steps]
    train_targets = data[1:len(train_states)+1]  # Predict next step
    
    reservoir.train_readout(train_states, train_targets.reshape(-1, 1))
    
    # Predict
    test_states = states[-prediction_steps:]
    predictions = reservoir.predict(test_states)
    
    return predictions.flatten()
```

### 2. Chaotic System Modeling

```python
def model_lorenz_attractor(reservoir, trajectory, duration=10.0):
    """
    Model Lorenz attractor dynamics.
    
    Args:
        reservoir: ParametricOscillatorReservoir
        trajectory: Lorenz trajectory data
        duration: Simulation duration
    
    Returns:
        modeled: Reservoir model of attractor
    """
    # Encode 3D trajectory
    encoded = reservoir.encode_input_sequence(trajectory)
    
    # Run reservoir
    states = reservoir.run_reservoir(encoded, duration_per_step=0.01)
    
    # Train readout on all three dimensions
    reservoir.train_readout(states[:-1], trajectory[1:])
    
    # Generate trajectory from reservoir
    initial_state = states[0]
    generated = [reservoir.predict(initial_state.reshape(1, -1))[0]]
    
    current = initial_state
    for _ in range(len(trajectory) - 1):
        # Evolve state (simplified - would need proper dynamics)
        next_state = states[len(generated)] if len(generated) < len(states) else current
        pred = reservoir.predict(next_state.reshape(1, -1))[0]
        generated.append(pred)
        current = next_state
    
    return np.array(generated)
```

## Energy Efficiency Analysis

```python
def analyze_energy_efficiency(n_neurons, operation_time):
    """
    Compare energy efficiency with digital implementations.
    
    Args:
        n_neurons: Number of neurons
        operation_time: Operation time (s)
    
    Returns:
        comparison: Energy comparison metrics
    """
    # Parametric oscillator parameters
    V_drive = 1.0  # V
    I_leak = 1e-9  # 1 nA leakage
    f_osc = 1e9  # 1 GHz
    
    # Energy per operation
    E_per_osc = 0.5 * V_drive**2 * I_leak / f_osc
    E_oscillator = n_neurons * E_per_osc * operation_time * f_osc
    
    # Digital implementation (MAC operation)
    E_per_mac_digital = 1e-12  # 1 pJ per MAC (optimistic)
    n_operations = n_neurons * n_neurons * operation_time * 100  # Assuming 100 Hz updates
    E_digital = n_operations * E_per_mac_digital
    
    # Energy ratio
    energy_ratio = E_digital / E_oscillator
    
    return {
        'oscillator_energy_J': E_oscillator,
        'digital_energy_J': E_digital,
        'energy_ratio': energy_ratio,
        'improvement_factor': f"{energy_ratio:.1e}x",
        'energy_per_neuron_pJ': E_per_osc * 1e12
    }
```

## Key Specifications

| Parameter | Value |
|-----------|-------|
| Operating frequency | 0.1-10 GHz |
| Energy per operation | < 1 fJ |
| Latency | < 1 ns |
| Coupling range | Local to global |
| Encoding dimension | 2×N (phase/amplitude) |
| Fan-in/Fan-out | Natural optical coupling |

## Advantages

1. **Ultra-low energy**: Sub-fJ per operation
2. **High speed**: GHz operation
3. **Natural connectivity**: Optical coupling provides fan-in/fan-out
4. **High-dimensional**: Phase and amplitude encoding
5. **Parallel processing**: Frequency comb enables parallelism

## Challenges

1. **Phase stability**: Requires phase-locked operation
2. **Coupling control**: Precise coupling matrix implementation
3. **Scalability**: Optical integration challenges
4. **Programming**: Weight programming in analog domain
5. **Noise**: Thermal and quantum noise effects

## References

- Kumar, M.S., & Ganesan, A. (2026). Neuromorphic Computing Based on Parametrically-Driven Oscillators and Frequency Combs. arXiv:2604.21861v1
- Mahboob, I., & Yamaguchi, H. (2008). Bit storage and bit flip operations in an electromechanical oscillator
- Shalóm, D.E., et al. (2015). Quantum memristors with superconducting circuits
- Pappas, D.P., et al. (2021). Coherent Ising machines with optical parametric oscillators

## Related Skills

- `coupled-oscillator-neural`: Coupled oscillator neural networks
- `neuromorphic-photonic`: Photonic neuromorphic systems
- `optical-reservoir`: Optical reservoir computing
- `frequency-comb-computing`: Frequency comb applications

## Activation Keywords

- parametric oscillator neuromorphic
- frequency comb computing
- coupled oscillator network
- optical parametric resonance
- ultra-fast neuromorphic
- phase-amplitude encoding
- parametric resonance computing