Noise-constrained switching times for <i>heteroclinic computing</i>

Neves, Fábio Schittler; Voit, M.; Timme, Marc

doi:10.1063/1.4977552

Cited by 14 publications

(15 citation statements)

References 31 publications

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“…Indeed, it has already been shown that the first higher‐order term in Equation (4) involves the local isochron curvature . Beyond the homoclinic orbits considered here, it is also exciting to consider oscillatory behavior originating from heteroclinic loops …”

Section: Discussionmentioning

confidence: 99%

Phase‐response curves of ion channel gating kinetics

Schleimer

Schreiber

2018

Math Methods in App Sciences

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Reduced models of single-neuron dynamics are widely used to characterize neuronal properties such as stimulus encoding and a cell's contribution to network function. Low-dimensional phase oscillator models based on phase-response curves, in particular, permit to capture the essence of a neuron's input-output dynamics at the level of spike times from high-dimensional conductance-based neuron models. In principle, these phase models allow for a direct translation of perturbations in the state variables (such as changes in specific ion channel conductances and kinetics) to shifts in the timing of spikes. Previous mathematical work, however, has predominantly focused on voltage and current perturbations and less on perturbations occurring in any of the involved kinetic equations of ionic channels. Here, we provide the means to mathematically relate properties in these variables to the phase-dependent voltage dynamics. Specifically, we derive the vector of phase-response curves near two different onset bifurcations: the physiologically prominent saddle node on invariant cycle bifurcation as well as the saddle-node loop bifurcation. We demonstrate that, locally around the saddle node, the tangent space of the isochrons is spanned by the strongly stable manifold of the saddle node. This finding enables us to quantitatively relate perturbations in the gating kinetics of ion channels to the resulting changes in spike timing. These results lay the methodological foundation for future quantitative analyses of the impact of specific biophysical channel properties on spike timing, including, but not limited to, channel noise. KEYWORDSconductance-based neuron model, gating kinetics, homoclinic orbits to a saddle node, isochrons, phase-response curves, saddle-node loop, saddle node on invariant cycle, semistable manifold, strongly stable manifold 8844

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Section: Discussionmentioning

confidence: 99%

Phase‐response curves of ion channel gating kinetics

Schleimer

Schreiber

2018

Math Methods in App Sciences

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“…More complex attractors consisting of networks of invariant sets in phase space have been proposed in the literature [41,60]. Such models found renewed interest in neuroscience [38,58] and other fields of research, as they provide a fundamental tool to describe dynamic processes occurring on transients that explore excitable connections.…”

Section: Network Attractorsmentioning

confidence: 99%

“…Both curves come out of the invariant square [−1, 1] 2 with their humps, that guarantees 9 intersections, i.e., 9 fixed points. Bottom left: depiction of the sufficient condition (41); the origin is the only fixed point and it is unstable. Bottom right: illustration of the sufficient condition (42).…”

Section: B2 Esn Maps In Two Dimensionsmentioning

confidence: 99%

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Interpreting Recurrent Neural Networks Behaviour via Excitable Network Attractors

2019

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Introduction: Machine learning provides fundamental tools both for scientific research and for the development of technologies with significant impact on society. It provides methods that facilitate the discovery of regularities in data and that give predictions without explicit knowledge of the rules governing a system. However, a price is paid for exploiting such flexibility: machine learning methods are typically black-boxes where it is difficult to fully understand what the machine is doing or how it is operating. This poses constraints on the applicability and explainability of such methods. Methods: Our research aims to open the black-box of recurrent neural networks, an important family of neural networks used for processing sequential data. We propose a novel methodology that provides a mechanistic interpretation of behaviour when solving a computational task. Our methodology uses mathematical constructs called excitable network attractors, which are invariant sets in phase space composed of stable attractors and excitable connections between them. Results and Discussion: As the behaviour of recurrent neural networks depends both on training and on inputs to the system, we introduce an algorithm to extract network attractors directly from the trajectory of a neural network while solving tasks. Simulations conducted on a controlled benchmark task confirm the relevance of these attractors for interpreting the behaviour of recurrent neural networks, at least for tasks that involve learning a finite number of stable states and transitions between them.as reservoir computing [32,33]. Echo state networks (ESNs) [21,30] constitute an important example of reservoir computing, where a recurrent layer (called a reservoir) is composed of a large number of neurons with randomly initialised connections that are not fine-tuned via gradient-based optimisation mechanisms. The main idea behind ESNs is to exploit the rich dynamics generated by the reservoir with an output layer, the read-out that is optimised to solve a specific task. Problem statement and research hypothesisThe high-dimensional and non-linear nature of RNNs complicates interpretability of their internal dynamics, which are characterised by complex, input-dependent spatio-temporal patterns of activity [47,55]. This poses constraints on understanding the behaviour of RNNs: they are usually viewed as black-boxes from which it is hard to extract useful knowledge about their inner workings. As highlighted by recent research efforts [10,24,40], similar interpretability issues affect many other machine learning methods. Furthermore, an increasing societal need to develop accountability and explainability of decision making by AI [17] is driving the development of methodologies for explaining the behaviour of such methods.Our aim in this paper is to develop effective models that capture the essential dynamical behaviour of RNNs on computational tasks as input-driven responses of a dynamical system, while neglecting microscopic details of the RNN dynamics in phase...

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“…In our case, the dynamic modes of the CPG and switching between them are the essential part of an entire computer. Switching between the CPG modes is akin to heteroclinic computing [ 19 , 20 ]. We are working with relatively small circuits and believe that the principals of their functioning are quite instructive and can be applied to larger networks [ 21 ].…”

Section: Introductionmentioning

confidence: 99%

A ‘reader’ unit of the chemical computer

Smelov¹,

Vanag²

2018

R. Soc. open sci.

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We suggest the main principals and functional units of the parallel chemical computer, namely, (i) a generator (which is a network of coupled oscillators) of oscillatory dynamic modes, (ii) a unit which is able to recognize these modes (a ‘reader’) and (iii) a decision-making unit, which analyses the current mode, compares it with the external signal and sends a command to the mode generator to switch it to the other dynamical regime. Three main methods of the functioning of the reader unit are suggested and tested computationally: (a) the polychronization method, which explores the differences between the phases of the generator oscillators; (b) the amplitude method which detects clusters of the generator and (c) the resonance method which is based on the resonances between the frequencies of the generator modes and the internal frequencies of the damped oscillations of the reader cells. Pro and contra of these methods have been analysed.

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Noise-constrained switching times for heteroclinic computing

Cited by 14 publications

References 31 publications

Phase‐response curves of ion channel gating kinetics

Phase‐response curves of ion channel gating kinetics

Interpreting Recurrent Neural Networks Behaviour via Excitable Network Attractors

A ‘reader’ unit of the chemical computer

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