Problem Description

You are given an m x n integer matrix, neurons.

Definitions

• A cell is a firing neuron if its value is 0.
• A cell is a non-firing neuron if its value is greater than 0.

For each cell in the matrix, you must count the number of firing neighbors among its 8 surrounding cells (horizontal, vertical, and diagonal). All updates to the matrix must be applied simultaneously, meaning the calculation for every cell's new state must be based entirely on the original state of the matrix before any updates are made.

Update Rules

Apply the following rules to update the matrix:

1. If a cell is currently firing (0) and exactly 3 of its neighbors are firing, its new value becomes 6.
2. If a cell is currently non-firing (value >0) and it has 0 or 1 firing neighbors, decrease its value by 2.
3. If a cell is currently non-firing (value >0) and it has more than 3 firing neighbors, decrease its value by 1.
4. A cell's value can never go below 0. If a decrease would result in a negative value, the new value should be 0.
5. In all other cases not covered by rules 1-3, the cell keeps its current value.

Task

Implement a function that takes the neurons matrix and updates its state according to the rules.

Follow-up

• First, solve the problem using a copied matrix to manage the simultaneous updates.
• Then, optimize the solution to use O(1) auxiliary space.