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Gradient modelling of memristive systems

Fulvio Forni, Rodolphe Sepulchre

TL;DR

This work develops a gradient-based framework for memristive systems by treating memconductance as a gradient operator on quadratic functionals under a memconductance-derived metric, extending gradient modeling from resistors to memristive elements operating on past-trajectory spaces. It shows that memristive elements can be characterized by a pair of energy-like quantities (dissipation and co-content) in trajectory space, and extends to memRC circuits and Hodgkin–Huxley-type neurons. The approach yields a decoupled two-layer view: a feedforward, fading-memory memconductance and a gradient, gradient-based memRC circuit. The framework offers a principled method for analysis and design of neuromorphic circuits and suggests new numerical approaches via trajectory-space zero-finding and operator-splitting.

Abstract

We introduce a gradient modeling framework for memristive systems. Our focus is on memristive systems as they appear in neurophysiology and neuromorphic systems. Revisiting the original definition of Chua, we regard memristive elements as gradient operators of quadratic functionals with respect to a metric determined by the memristance. We explore the consequences of gradient properties for the analysis and design of neuromorphic circuits.

Gradient modelling of memristive systems

TL;DR

This work develops a gradient-based framework for memristive systems by treating memconductance as a gradient operator on quadratic functionals under a memconductance-derived metric, extending gradient modeling from resistors to memristive elements operating on past-trajectory spaces. It shows that memristive elements can be characterized by a pair of energy-like quantities (dissipation and co-content) in trajectory space, and extends to memRC circuits and Hodgkin–Huxley-type neurons. The approach yields a decoupled two-layer view: a feedforward, fading-memory memconductance and a gradient, gradient-based memRC circuit. The framework offers a principled method for analysis and design of neuromorphic circuits and suggests new numerical approaches via trajectory-space zero-finding and operator-splitting.

Abstract

We introduce a gradient modeling framework for memristive systems. Our focus is on memristive systems as they appear in neurophysiology and neuromorphic systems. Revisiting the original definition of Chua, we regard memristive elements as gradient operators of quadratic functionals with respect to a metric determined by the memristance. We explore the consequences of gradient properties for the analysis and design of neuromorphic circuits.

Paper Structure

This paper contains 14 sections, 2 theorems, 78 equations, 3 figures.

Table of Contents

  1. introduction
  2. Memristive systems
  3. Definition and examples
  4. Fading memory
  5. Dissipativity
  6. Gradient models of memristive elements
  7. Gradient models of resistive elements
  8. From resistive to memristive elements
  9. Gradient memristive networks
  10. Gradient memRC circuits
  11. Gradient modeling of RC circuits
  12. From RC to memRC circuits
  13. Hodgkin-Huxley model
  14. Discussion

Key Result

Theorem 1

Given the Riemannian manifold $({\mathbb R}, \frac{1}{g})$, consider $\mathcal{D}_{\frac{1}{g}} = v^2$ and $\mathcal{E}_{\frac{1}{g}} = \frac{1}{2}v^2$. Then, the current/voltage relationship of the nonlinear resistor eq:nonlinear_resistor at any $v$ satisfies $\lrcorner$

Figures (3)

  • Figure 1: The 'linearized' area element due to the metric $\frac{1}{g}$.
  • Figure 2: Conductance-based neuron equivalent circuit.
  • Figure 3: Hodgkin-Huxley spike. Top: ODE solution via nmumerical integration. Bottom|left: ODE solution via alternate iteration. Bottom|right: external current.

Theorems & Definitions (9)

  • Example 1: Potassium current model
  • Example 2: Synaptic current model
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Example 3
  • Example 4