Project: s41467-024-45772-4 | Function: solve_ivp_odes (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:** Solves the system of ordinary differential equations for embryonic pattern formation using Heaviside functions and Euler method integration

**KEY TERMS:** {
  "b_new": "BMP4 concentration B_i",
  "c_new": "calcium activity C_i",
  "v_new": "cVG1 concentration V_i",
  "a_new": "streak cell identity A_i",
  "H_c": "Heaviside function H(C_i - α)",
  "H_v_b": "Heaviside function H(β_V - B_i)",
  "H_c_b": "Heaviside function for streak identity",
  "dt": "time step Δt",
  "dx": "spatial step Δx"
}

**CONTEXT CODE:** import numpy as np
from scipy import integrate
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from copy import deepcopy


def calcium_periodic_boundary(c, number_of_cells, cell_idx):
    
    cell_idx_plus_one = (cell_idx + 1) % number_of_cells
    cell_idx_minus_one = (cell_idx - 1) % number_of_cells
    
    c_bar = 0.5 * (c[cell_idx_plus_one] + c[cell_idx_minus_one])
    
    return c_bar
    
def calcium_no_flux_boundary(c, number_of_cells, cell_idx):
    # MASKED: Implement no-flux boundary conditions for calcium diffusion in spatial domain implementation
    raise NotImplementedError("This function is masked for testing purposes.")
    
def solve_ivp_odes(number_of_cells, number_of_timepoints, dt, dx, a0, b0, c0, v0, calcium_boundary_condition, params, sample_rate):
    # MASKED: Solve system of ODEs for embryonic pattern formation using Heaviside functions and Euler method integration implementation
    raise NotImplementedError("This function is masked for testing purposes.")

def solve_ivp_odes_Hill(number_of_cells, number_of_timepoints, dt, dx, a0, b0, c0, v0, calcium_boundary_condition, params, sample_rate):
    # MASKED: Alternative implementation using Hill functions instead of Heaviside functions for smoother regulation in embryonic pattern formation implementation
    raise NotImplementedError("This function is masked for testing purposes.")



Complete the function implementation replacing "raise NotImplementedError".