Unit-5: Neuro Dynamics

Neuro Dynamics: Dynamical Systems, Stability of Equilibrium States, Attractors, Neuro Dynamical Models, Manipulation of Attractors as a Recurrent Network Paradigm Hopfield Models – Hopfield Models, restricted boltzmen machine.

Neuro Dynamics refers to the study of how the state of a neural network evolves over time according to specific update rules or differential equations. It focuses on the temporal behaviour, stability, and trajectory of neural activations until the network reaches a steady state or attractor.

In simple words:

Neuro Dynamics = Time-evolution of neuron states + stability of the network’s behaviour.

It treats the neural network as a dynamical system rather than a static input–output model.

What is Neuro Dynamics?

In Neural Networks, Neuro Dynamics describes:

  • How neuron activations change step-by-step or continuously
  • How the network moves in its state space
  • How it settles  into stable patterns (memories or attractors)
  • How noise, learning, or weight changes modify trajectories
Mathematically, it is represented as:
  • Continuous-time:        
                                
  • Discrete-time:          
                         Here ‘x’ represents the network's state vector.

Purpose of Neuro Dynamics in Neural Network

Understanding Stability of Neural Systems

    • Determine if the network will converge or diverge
    • Identify stable equilibrium points (memory states)

Designing Memory-Based Networks

    • Enables models like Hopfield Networks, Boltzmann Machines
    • Networks can recall stored patterns using attractor dynamics

Error Correction & Pattern Completion

    • In attractor networks, even noisy or incomplete inputs converge to correct stored patterns.

 Modelling Biological Neural Behaviour

    • Helps simulate rhythms, oscillations, or neural population dynamics
      (e.g., Wilson–Cowan, cortical circuits).

Analyzing Learning & Weight Updates

    • Dynamical systems theory helps examine how weights evolve.
    • Prevents chaotic behaviour or instability during training.

Optimization & Energy Minimization

    • Many neurodynamic networks work by minimizing an energy function.
      → Used for solving optimization problems.

Applications of Neuro Dynamics in Neural Networks

            Associative Memory Systems

    • Hopfield Networks
    • Ability to recall stored patterns
    • Content-addressable memory (CAM)

            Pattern Recognition & Completion

    • Network fills missing data
    • Robust to noisy inputs

Optimization Problems

Neurodynamic models (e.g., Hopfield networks) can solve:

    • Travelling Salesman Problem (TSP)
    • Assignment problems
    • Graph optimization
    • Combinatorial optimization

Brain-Like Computation & Cognitive Modelling

    • Modelling decision making
    • Neural oscillations
    • Attractor-based cognitive tasks (working memory)

Recurrent Neural Networks

    • Neuro Dynamics explains convergence & stability in RNNs, LSTMs, Echo-State Networks.

Energy-Based Models

    • Boltzmann Machines
    • Restricted Boltzmann Machines
    • Deep Belief Networks

Control Systems & Robotics

    • Neuro Dynamic controllers
    • Adaptive control
    • Real-time navigation and motor control

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