Enter Model Inputs
This calculator uses a one-state mean-field neural dynamics model. The full page stays single-column, while the input grid becomes three columns on large screens.
Formula Used
1. Effective field
hk = w·xk + I + F·sin(2πνtk) + εk
2. Nonlinear activation
ak = tanh(β·hk)
3. Euler state update
xk+1 = xk + Δt[ -λxk + ak ]
4. Local stability check
m = 1 + Δt[ -λ + βw·sech²(β(wx* + I)) ]
5. Energy proxy
V(x) = 0.5λx² - Ix - ln(cosh(β(wx + I))) / β
The calculator models a compact neural dynamical system using a mean-field approximation. Decay pulls the state downward, feedback weight amplifies recurrence, and the tanh activation keeps the response bounded. Periodic forcing and deterministic noise help explore resonance, disturbance sensitivity, and damping behavior.
The stability margin is derived from the local derivative of the discrete map. Values with |m| < 1 indicate local convergence near the final state. The oscillation measures summarize late-stage fluctuations, while the energy proxy gives a Lyapunov-like signal for comparing scenarios.
How to Use This Calculator
- Enter the initial neural state and choose the average feedback weight.
- Set the external input, decay rate, and activation gain.
- Choose the time step and total number of iterations.
- Add a target state to measure final tracking error.
- Use forcing amplitude and frequency to test periodic excitation.
- Use noise amplitude to inspect robustness under disturbances.
- Press Calculate Dynamics to show results above the form.
- Review the graph, stability metrics, and downloadable timeline.
Example Data Table
The table below uses the built-in default example values. It shows selected timeline snapshots from the simulated trajectory.
| Step | Time | State | Field | Activation | Target Error |
|---|---|---|---|---|---|
| 0 | 0.00 | 0.2000 | 0.5500 | 0.7329 | 0.6000 |
| 10 | 1.00 | 0.8453 | 1.4717 | 0.9867 | 0.0453 |
| 20 | 2.00 | 1.2613 | 1.8702 | 0.9965 | 0.4613 |
| 30 | 3.00 | 1.5002 | 2.1812 | 0.9988 | 0.7002 |
| 40 | 4.00 | 1.6372 | 2.6187 | 0.9997 | 0.8372 |
| 50 | 5.00 | 1.7152 | 2.5977 | 0.9997 | 0.9152 |
| 60 | 6.00 | 1.7594 | 2.4906 | 0.9996 | 0.9594 |
FAQs
1. What does this calculator measure?
It simulates how a neural state evolves over time under recurrence, decay, external drive, forcing, and bounded nonlinear activation. It also estimates stability, convergence, oscillation, energy behavior, and target-tracking error from the generated trajectory.
2. Why is the tanh activation used?
The tanh function is smooth, nonlinear, and bounded between −1 and 1. That makes it useful for illustrating saturation, damping, and stable feedback behavior in compact neural dynamics models.
3. What does the stability margin mean?
It is derived from the local slope of the discrete update near the final state. Positive margin with |m| below 1 suggests local convergence, while negative margin warns that nearby perturbations may grow.
4. What is the oscillation strength?
Oscillation strength summarizes the late-stage variation of the state relative to its recent average. Higher values indicate persistent ripples, cycles, or unstable settling instead of smooth convergence.
5. Why include forcing and noise?
They help test robustness. Forcing reveals resonance-like responses to periodic input, while noise shows whether the system damps disturbances or amplifies them under the selected feedback and gain values.
6. What does the convergence score show?
It compares the size of the final update with the initial update. A higher score means the motion slowed substantially by the end, which usually signals stronger settling.
7. Is this a full multi-neuron simulator?
No. It is a one-state mean-field approximation. That keeps the tool interpretable and fast, while still capturing key ideas from recurrent neural dynamics such as feedback, saturation, damping, and local stability.
8. When should I reduce the time step?
Use a smaller time step when the graph looks jagged, the gain is large, or the state changes too abruptly. Smaller steps usually improve numerical smoothness and stability estimation.