Project: s41467-025-58231-5 | Function: run (python)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Implements the main Neurosa optimization algorithm that mimics biological neural networks to solve maximum cut problems. Iteratively selects neurons randomly, updates their states based on membrane potentials, and tracks the best solution found during the stochastic optimization process.

**KEY TERMS:** {
  "MAX_ITER": "maximum optimization iterations",
  "update_neuron": "neuromorphic state update",
  "best_maxcuts": "optimal solution tracking",
  "thld": "threshold parameter"
}

**CONTEXT CODE:** 
Author
-------
Zihao Chen 08/01/2024

License
-------
This work is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License.
To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/4.0/.

You are free to:
- Share — copy and redistribute the material in any medium or format
- Adapt — remix, transform, and build upon the material

Under the following terms:
- Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- NonCommercial — You may not use the material for commercial purposes.

No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
"""

import numpy as np
import math
import random

class Neurosa:
    def __init__(self, Q, max_iter=1e8, thld_max=8e4, thld_delta=2e-3):
        self.Q = Q
        self.N, self.M = Q.shape
        self.MAX_ITER = max_iter
        self.thld_max = thld_max
        self.thld_delta = thld_delta
        self.DTYPE = np.float32
        
        self.ds_p = np.zeros((self.N,), dtype=self.DTYPE)
        self.ds_n = np.zeros_like(self.ds_p, dtype=self.DTYPE)
        self.s_p = np.ones_like(self.ds_p, dtype=self.DTYPE)
        self.s_n = np.zeros_like(self.ds_p, dtype=self.DTYPE)
        self.vmem = np.matmul(Q, (self.s_p - self.s_n))
        
        self.num_spikes = 0
        self.best_maxcuts = 0
        self.maxcuts = 0
        self.iter = 0
        self.thld_iter = 0
        self.thld = 1
        
        self.v = (self.s_p - self.s_n)
        self.curr_cuts = np.sum(np.sum(np.multiply(Q, 1 - np.outer(self.v, self.v)))) / 4
    
    def run(self):
        """
        Implements the main Neurosa optimization algorithm that mimics biological neural networks to solve maximum cut problems.

        This function iteratively selects neurons randomly, updates their states based on membrane potentials,
        and tracks the best solution found during the stochastic optimization process. The neuromorphic approach
        uses biologically-inspired mechanisms to escape local optima in combinatorial optimization problems.

        The algorithm is particularly beneficial when minimal gradient information is available and the optimization
        has to resort to brute force random search, as described in the paper for solving complex combinatorial
        optimization problems (COPs) using neuromorphic computing principles.

        Key algorithmic elements:
        - Iterative optimization through random neuron selection
        - Neuromorphic state updates via membrane potential calculations
        - Stochastic threshold updates to guide optimization process
        - Progress tracking of best solutions found during search

        Paper reference: "Because the SA algorithm itself can be slow, the benefits of NeuroSA compared to
                        other polynomial-time COP solvers (like the GW algorithm) lie in the regime where
                        minimal gradient information is available to guide the optimization process, and the
                        algorithm has to resort to a brute force random search"

        Formula: Neuromorphic optimization through stochastic neuron updates
        Implementation: Main optimization loop with MAX_ITER iterations, random neuron selection,
                       threshold updates, and solution tracking
        """
        # MASKED: Neuromorphic optimization algorithm implementation
        raise NotImplementedError("This function is masked for testing purposes. The function implements the main Neurosa optimization algorithm that mimics biological neural networks to solve maximum cut problems, iteratively selecting neurons randomly, updating their states based on membrane potentials, and tracking the best solution found during the stochastic optimization process.")
    
    def update_neuron(self, p):
        """
        Updates individual neuron states using biologically-inspired spiking mechanisms.

        This function computes noise threshold, determines spiking conditions based on membrane potential,
        and updates neuron states (positive and negative spikes) along with the overall cut count for the
        optimization problem. The method implements neuromorphic computing principles to simulate biological
        neural network behavior for combinatorial optimization.

        The spiking condition follows the formula from the paper, where neurons spike when the stochastic
        threshold condition is met, enabling the algorithm to explore different solution spaces through
        biologically-motivated state transitions.

        Args:
            p (int): Index of the neuron to update

        Key algorithmic elements:
        - Stochastic computation of noise threshold using logarithmic functions
        - Conditional logic for spike determination based on membrane potential
        - State updates for positive and negative neuron components
        - Mathematical operations for cut count maintenance and optimization tracking

        Paper reference: "In the initial phase of hybrid approach, the optimization endeavor follows the
                        steepest gradient descent and performs parallel updates on neuron states resembling SDP"

        Formula: Spike condition: (noisethld * (s_p - s_n) - vmem) * (s_p - s_n) < 0
        Implementation: Neuromorphic state update with stochastic threshold computation,
                       spike detection, and solution quality tracking
        """
        # MASKED: Biologically-inspired neuron state update implementation
        raise NotImplementedError("This function is masked for testing purposes. The function updates individual neuron states using biologically-inspired spiking mechanisms, computing noise threshold, determining spiking conditions based on membrane potential, and updating neuron states along with the overall cut count for the optimization problem.")


Complete the function implementation replacing "raise NotImplementedError".