Fractional-Order Neural Network Dynamics

Fractional-order neural networks extend classical integer-order models by incorporating derivatives of non-integer order, thereby capturing long-range temporal correlations and memory effects inherent in many real-world processes. By replacing standard differential operators with fractional counterparts, these networks exhibit richer dynamic behaviours, including intermediate regimes between stability and chaos, enhanced adaptability to time-varying inputs and finer control of convergence rates. Such features have proven valuable in fields as diverse as signal processing, pattern recognition and secure communications, where fractional dynamics afford more accurate characterisation of anomalous diffusion, sub- and super-diffusive phenomena, and fading memory. The interplay between network structure, fractional order and time delays gives rise to a spectrum of synchronisation and stability properties that can be tuned via appropriate control laws. Recent advances have focused on devising systematic methods for analysing global stability in the sense of Mittag-Leffler, designing hybrid or adaptive controllers, and exploring both continuous- and discrete-time formulations. Together, these developments underscore the global significance of fractional-order neural networks as versatile tools for modelling complex systems with hereditary properties, and point towards practical applications in neuromorphic computing, sensor networks and biological modelling.

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Research from all publishers

One strand of research has introduced hybrid control schemes for projective synchronisation of memristive bidirectional associative memory networks with mixed time delays. By combining open-loop and adaptive state feedback, and employing Lyapunov–Krasovskii functionals alongside Barbalat’s lemma, researchers established sufficient criteria for global projective lag, complete and anti-synchronisation. Numerical examples confirmed the efficacy of these controllers in complex fractional-order models. Another line of work has addressed adaptive synchronisation of coupled fractional-order networks via quantised output control. Novel adaptive laws incorporating logarithmic quantisation were devised to stabilise the fractional-order error dynamics under output coupling, with stability proofs grounded in fractional Lyapunov functionals and linear matrix inequalities. Simulations demonstrated robust synchronisation even under communication constraints. In the discrete-time setting, fixed-point techniques applied to semilinear fractional difference equations have yielded Mittag-Leffler stability conditions for neural networks with and without delays. By constructing equivalent sum representations using discrete Mittag-Leffler functions and contraction mappings, researchers proved finite-time stability and attractivity results, highlighting the versatility of fixed-point approaches for discrete fractional systems.

Fractional-Order Neural Network Dynamics publication trend

The graph below shows the total number of articles in fractional-order neural network dynamics across all publications each year (not limited to Nature Index journals).

Technical terms

Fractional derivative: A generalisation of the standard derivative to non-integer orders, capturing memory and hereditary effects in dynamical systems.

Memristive neural network: A neural model incorporating memristors—elements whose resistance depends on past current—enabling synaptic plasticity and memory.

Projective synchronisation: A form of synchronisation where the response system’s state tracks the drive system up to a constant scaling factor.

Lyapunov–Krasovskii functional: An extension of Lyapunov functions to systems with delays, used to derive stability conditions via integral terms.

Linear matrix inequality (LMI): A convex constraint on matrix variables, widely used to express and solve stability and controller-design conditions.

References

1. Hybrid Control Scheme for Projective Lag Synchronization of Riemann–Liouville Sense Fractional Order Memristive BAM NeuralNetworks with Mixed Delays. Mathematics (2019).
2. Adaptive Synchronization of Fractional-Order Output-Coupling Neural Networks via Quantized Output Control. IEEE Transactions on Neural Networks and Learning Systems (2021).
3. Mittag-Leffler stability analysis of fractional discrete-time neural networks via fixed point technique. Nonlinear Analysis Modelling and Control (2019).