Project: s41467-025-58741-2 | Function: stochastic_model_noise (python)

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**DESCRIPTION:** Implementation of Ornstein-Uhlenbeck process with Gaussian noise component only, providing analytical solution for the stochastic differential equation without jumps

**KEY TERMS:** {
  "sigma_t": "time-dependent noise standard deviation σₑ",
  "np.random.normal": "Gaussian noise εₜ",
  "g_tilde": "equilibrium state g̃",
  "theta": "reversion speed θ"
}

**CONTEXT CODE:** 
Vin_list=[x]*n

##Time list
delta_t=[dt]*n
timesteps= len(delta_t)
time_list=[]
time_list.append(0)

for t in range(1,len(delta_t)):
    s = time_list[t-1] + delta_t[t]
    time_list.append(s)
    
#%% DETERMINISTIC MODEL

#iterative solution

def deterministic_model(time_list, kp0, kd0, eta_p, eta_d, G_0, G_min, G_max):
    """
    Implements the deterministic component of the Ornstein-Uhlenbeck process for neuromorphic nanowire networks.

    This function models the deterministic dynamics of memristive conductance evolution based on the kinetic
    rate equations for potentiation (kp) and depression (kd) processes. The implementation follows the
    mathematical framework described in the paper for self-organizing neuromorphic networks.

    Args:
        time_list: Array of time points for simulation
        kp0: Base potentiation rate constant
        kd0: Base depression rate constant
        eta_p: Potentiation voltage sensitivity parameter
        eta_d: Depression voltage sensitivity parameter
        G_0: Initial conductance value
        G_min: Minimum conductance boundary
        G_max: Maximum conductance boundary

    Returns:
        tuple: (g, G) normalized and absolute conductance arrays

    Paper reference: "The deterministic part of the model captures the average evolution of the system"

    Formula: g[i] = g_tilde*(1-exp(-theta*delta_t)) + g[i-1]*exp(-theta*delta_t)
    Implementation: Exponential decay kinetics, boundary conditions, and conductance normalization
    """
    # MASKED: Deterministic Ornstein-Uhlenbeck process implementation for neuromorphic conductance modeling
    raise NotImplementedError("This function is masked for testing purposes. The function implements the deterministic component of the Ornstein-Uhlenbeck process for neuromorphic nanowire networks, modeling memristive conductance evolution using kinetic rate equations.")

#%% STOCHASTIC MODEL (DETERMINISTIC + NOISE)

#iterative solution

def stochastic_model_noise(time_list, kp0, kd0, eta_p, eta_d, G_0, G_min, G_max, sigma):
    """
    Implements the Ornstein-Uhlenbeck process with Gaussian noise for neuromorphic nanowire networks.

    This function extends the deterministic model by incorporating Gaussian noise to capture stochastic
    fluctuations in the conductance evolution. The implementation follows the analytical solution of the
    stochastic differential equation with time-varying variance as described in the paper.

    Args:
        time_list: Array of time points for simulation
        kp0: Base potentiation rate constant
        kd0: Base depression rate constant
        eta_p: Potentiation voltage sensitivity parameter
        eta_d: Depression voltage sensitivity parameter
        G_0: Initial conductance value
        G_min: Minimum conductance boundary
        G_max: Maximum conductance boundary
        sigma: Noise amplitude parameter

    Returns:
        tuple: (g, G) stochastic normalized and absolute conductance arrays

    Paper reference: "We add Gaussian noise to the deterministic model to account for thermal fluctuations"

    Formula: g[i] = deterministic_term + sigma_t*N(0,1) where sigma_t = sqrt(((1-exp(-2*theta*dt))*sigma^2)/(2*theta))
    Implementation: Ornstein-Uhlenbeck analytical solution, time-dependent noise variance, and stochastic integration
    """
    # MASKED: Ornstein-Uhlenbeck process with Gaussian noise for stochastic neuromorphic modeling
    raise NotImplementedError("This function is masked for testing purposes. The function implements the Ornstein-Uhlenbeck process with Gaussian noise for neuromorphic nanowire networks, modeling stochastic conductance fluctuations using analytical SDE solutions.")

#%% STOCHASTIC MODEL (DETERMINISTIC + NOISE + JUMPS)

#Euler Mayurama solution

def stochastic_model_noise_jumps(time_list, kp0, kd0, eta_p, eta_d, G_0, G_min, G_max, sigma, A_exp, A_coeff, A_min, A_max):
    """
    Implements the complete Ornstein-Uhlenbeck jump-diffusion process for neuromorphic nanowire networks.

    This function implements the full stochastic model including both Gaussian noise and Poisson jump processes
    using the Euler-Maruyama numerical method. The jump process follows a power-law amplitude distribution
    and homogeneous Poisson arrival times, capturing the discrete switching events in memristive devices.

    Args:
        time_list: Array of time points for simulation
        kp0: Base potentiation rate constant
        kd0: Base depression rate constant
        eta_p: Potentiation voltage sensitivity parameter
        eta_d: Depression voltage sensitivity parameter
        G_0: Initial conductance value
        G_min: Minimum conductance boundary
        G_max: Maximum conductance boundary
        sigma: Gaussian noise amplitude parameter
        A_exp: Power-law exponent for jump amplitude distribution
        A_coeff: Power-law coefficient for jump amplitude distribution
        A_min: Minimum jump amplitude boundary
        A_max: Maximum jump amplitude boundary

    Returns:
        tuple: (g, G) complete stochastic normalized and absolute conductance arrays

    Paper reference: "We model the jumps using a Poisson process with power-law distributed amplitudes"

    Formula: dg = theta*(g_tilde-g)*dt + sigma*dW + J*dN where J~power-law, N~Poisson(E*dt)
    Implementation: Euler-Maruyama integration, Poisson jump sampling, power-law amplitude generation,
                   and bias-dependent jump direction determination
    """
    # MASKED: Complete Ornstein-Uhlenbeck jump-diffusion process for comprehensive neuromorphic modeling
    raise NotImplementedError("This function is masked for testing purposes. The function implements the complete Ornstein-Uhlenbeck jump-diffusion process for neuromorphic nanowire networks, modeling both continuous fluctuations and discrete switching events using Euler-Maruyama numerical integration.")


#%%% MODELING CONDUCTANCE TIME TRACE

#set initial condition
G_0 = 1.23364E-5

# Main execution block - only run when script is executed directly, not when imported
if __name__ == "__main__":
    #model_deterministic_2
    g_list_deterministic_model, G_list_deterministic_model = deterministic_model(time_list, kp0, kd0, eta_p, eta_d, G_0, G_min, G_max)
    #model_deterministic_stochastic_iterative
    g_list_stochastic_noise_model, G_list_stochastic_noise_model = stochastic_model_noise(time_list, kp0, kd0, eta_p, eta_d, G_0, G_min, G_max, sigma)
    #model_deterministic_stochastic_jumps_Euler_2
    g_list_stochastic_noise_jumps_model, G_list_stochastic_noise_jumps_model = stochastic_model_noise_jumps(time_list, kp0, kd0, eta_p, eta_d, G_0, G_min, G_max, sigma, A_exp, A_coeff, A_min, A_max)

#%% PLOTS

# Main execution block - only run when script is executed directly, not when imported
if __name__ == "__main__":
    #Plot G
    plt.figure(10,figsize=(10,7))
    #plt.title(title)
    plt.plot(time_list, G_list_deterministic_model, 'g', label='Deterministic model', linewidth=2.2)
    plt.plot(time_list, G_list_stochastic_noise_model, 'k', label='Stochastic model (deterministic + noise)', linewidth=2.2)
    plt.plot(time_list, G_list_stochastic_noise_jumps_model, 'r', label='Stochastic model (deterministic + noise + jumps)', linewidth=2.2)
    plt.legend(fontsize=20)
    plt.grid()
    plt.xlabel('time [s]', fontsize=20)
    plt.ylabel('G [S]', fontsize=20)
    plt.xticks(fontsize=20)
    plt.yticks(fontsize=20)


Complete the function implementation replacing "raise NotImplementedError".