# Non-digital computing: neuromorphic applications and mathematical theory

##### An old quest taking a fresh turn

The world-transforming powers of Turing / digital computing need no explanation. Yet, alternative paradigms of “computing” have been proposed in waves and cycles since the beginnings of the computing era, some short-lived, others more enduring. The graphics below gives an impression of the diversity in names, ideas, communities (and PR wizards, too).

*The diversity of non-digital computing approaches, highlighted by some of the proposed namings.*

Some niches and quantum computing excepted, academic impact and industrial involvement has been ephemeral.

This has started to change. Two major motivations to re-inspect and re-invest in non-digital computing are that

- physical ceilings for CMOS based digital microchip technologies are coming into sight (“End of Moore’s Law”),
- the ever-growing energy hunger of digital infrastructure is becoming insupportable.

Specifically, considerable and considerably fast growing efforts are now being spent on innovative “neuromorphic” (wide sense) approaches.

##### Optical and neuromorphic reservoir computing at MINDS

Due to its leading role in reservoir computing, MINDS has naturally been pulled first into optical, then additionally into neuromorphic hardware manifestations of reservoir computing, and finally, into nonstandard hardware research in general. Besides EU project reviewing and holding tutorials, we co-organized

- the
workshop
*Technology and Architectures Development for Brain Inspired Integrated Circuits*(at ESSCIRC - ESSDERC 2016), - the
conference
*Dynamical Systems and Brain Inspired Computing*(ULB Brussels, June 2017), - the
workshop
on
*Dynamical Systems and Brain-inspired Information Processing*(Oct 2017, Univ. Konstanz, Germany), - the conference
*Cognitive Computing: Merging Concepts with Hardware*, a flagship event of the Volkswagen Foundation (Dec 2018).

In our own research, we are currently busy realizing reservoir
computing learning on the analog spiking
DYNAP-se microprocessors, within
the EU H2020 project NeurRAM3
(2016-2019). The challenges lie in the
combination of extremely low numerical precision,
stochastic dynamics, device mismatch and unobservability
of core physical system parameters. **Image:** Shaping and stabilizing the dynamics of a spiking recurrent neural network, in a not quite trivial multi-timescale task. Solved challenge: adaptable synaptic weights only had 1.6 bit precision. Green: input signal, red: target output, blue: network output signal.

##### Mathematical foundations of non-digital computing

For a computer scientist, the answer to the question, “what is computing”, is given with magnificent authority by the formal model of Turing computability.

But, Turing computability hardly looks like the right model for understanding brains or other unconventional physical materials and media. Modelers must pick their tools from a scattered range of options, including

- dynamical systems theory (with dozens of subfields),
- stochastic processes (with dozens of subfields),
- statistical physics (with dozens of subfields),
- analytical and algebraic geometry and topology (with dozens of subfields).

A coherent mathematical foundation for non-digital computing, which
could step into the supremely unifying role that Turing computability
has served for digital computing, is missing. In a tutorial talk
delivered at the *Beyond von-Neumann Computing: Computing meets
Nanophysics*
Leopoldina Symposium (2017),
Herbert Jaeger explores the history of “computing” concepts and works
out challenges for future formalizations
(slides).

Research at MINDS has always been inclined towards mathematical theory. Our main contributions to mathematical formalizations of information processing systems (wide sense) are

*observable operator models*and*conceptors*, two mathematical modeling paradigms which clarify fundamental properties of stochastic and high-dimensional nonlinear dynamics, respectively; and have immediate bearings on the design of novel learning algorithms;- a unification of existing concepts of attractors in input-driven dynamical systems, relevant for “computing” because attractor concepts are often invoked as a link to explain discrete symbols in continuous dynamics (G. Manjunath, H. Jaeger (2014): The Dynamics of Random Difference Equations is Remodeled by Closed Relations. SIAM Journal on Mathematical Analysis 46(1), 459-483 (pdf);
- in a recently started effort, Herbert Jaeger is at present working out a definition of “probability” which differs from the standard Kolmogorov axiomatization and would allow to understand probability as being hierarchically information-encoding (stay tuned).