Neuromorphic Control
NNeuromorphic Control
Designing multiscale mixed-feedback systems
Luka Ribar and Rodolphe SepulchreNovember 10, 2020
Neuromorphic electronic engineering takes inspiration from the biological organizationof nervous systems to rethink the technology of computing, sensing, and actuating. It startedthree decades ago with the realization by Carver Mead, a pioneer of VLSI technology, that theoperation of a conventional transistor in the analog regime closely resembles the biophysicaloperation of a neuron [1]. Mead envisioned a novel generation of electronic circuits that wouldoperate far more efficiently than conventional VLSI technology and would allow for a newgeneration of biologically inspired sensing devices. Three decades later, active vision has becomea technological reality [2], [3] and neuromorphic computing has emerged as a promising avenueto reduce the energy requirements of digital computers [4]–[7].Neuromorphic circuit architectures call for new computing, signal processing and controlparadigms. A most pressing challenge of neuromorphic circuit design is to cope with the transistor mismatch [8], [9], that is, the uncertainty that comes with operating in a mixedanalog-digital regime that utilizes large ensembles of tiny devices with significant variability.The transistor mismatch has been a main hurdle to the development of the field over the lasttwo decades [10]. For control engineers, the challenge is certainly reminiscent of the uncertaintybottleneck faced by the engineers of Bell Laboratories in the early days of long distance signaltransmission, eventually solved by Black’s invention of the negative feedback amplifier.The new uncertainty bottleneck of neuromorphic engineering calls for novel controlprinciples. Neuromorphic control aims at taking inspiration from biology to address thischallenge. Nervous systems do cope with the uncertainty and variability of their components.They are able to maintain robust and flexible function over an impressive span of spatial andtemporal scales. Understanding what are the control mechanisms that allow the regulation ofthe robust global behavior through the cumulative effect of control actions at smaller scalescould turn the variability of transistors into a feature rather than a hurdle. More generally, thegrowing understanding of how biology exploits variability could provide new inspiration forthe increasingly multiscale nature of engineering design [11] (see ”Multiscale in biology andengineering”). 1 a r X i v : . [ ee ss . S Y ] N ov control principle at the heart of biological systems is the prevalence of concurrentpositive and negative feedback pathways, acting in different temporal and spatial scales (see”Positive and negative feedback”). This mixed-feedback organization allows neural systems tocolocalize memory and processing capabilities, generating discrete events with memory whileat the same time continuously regulating their internal properties to allow for learning andadaption. In nervous systems, tuning the mixed feedback gains of neuronal architectures isprimarily achieved by neuromodulation . Neuromodulation is a general term that encompasses arealm of neurochemical processes that control the electrical activity of neurons. These regulatoryprocesses allow sensory networks to dynamically adjust their information processing capabilities[12], while providing the flexibility and robustness of biological clocks underlying repeatedactions such as movement or breathing [13], [14]. The capability of neuromodulatory actions totarget and modify local parts of neural networks provides the basis of the biological control ofnervous systems across scales [11], [15].The aim of this article is to introduce the methodology for designing and controlling mixedfeedback systems neuromorphically , that is, by mimicking core neuromodulation mechanismsto the operation of neuromorphic circuits. It presents a simple circuit component that acts as anelementary feedback element providing localized positive or negative feedback. The parallelinterconnection of such elements is sufficient to capture the feedback structure present inbiological neurons, and thus presents an avenue for studying the principles of neuromorphiccontrol. This interconnection structure augments the circuit with a simple control methodologythrough shaping its input-output current-voltage (I-V) characteristic.The article is primarily based on the PhD thesis [16] and the recent work on neuromorphicdesign and control presented in [17], [18]. It is grounded in recent work aimed at understandingneuronal behaviors as feedback control systems and neuromodulation as controller design. Theinterested reader is referred to several review articles [11], [15], [19], and to the technicalcontributions [20]–[26].The rest of the article is organized as follows. ”Excitability: a mixed feedback principle”reviews the key modeling principles of biological neurons in the form of excitable electricalcircuits. ”Synthesis and control of neuromorphic circuits” then introduces a novel circuitarchitecture amenable to control design and hardware realization while retaining the core mixedfeedback loop modules of biological neuronal networks. The proposed architecture uses thetraditional neuromorphic circuits introduced by Carver Mead. They are designed by a graphicalinput-output methodology reminiscent of loop shaping: it is the shape of the I-V curve of thecircuit in distinct timescales that determines the closed-loop behavior, and each element of thecircuit can be regarded as shaping a particular I-V curve. ”Why control the excitability of a2euron?” then discusses the significance of the single neuron neuromodulatory design for thedesign of neuronal networks. As a main illustration, the article focuses on the concept of ”Nodalcontrol of a network”, which shows how modulating the excitability properties of the nodesof a network provides a versatile control principle to continuously reconfigure the functional topology of the network without altering its connectivity. This basic principle is illustrated on afive neuron network inspired from the crustacean stomatogastric ganglion (STG) that has servedas a central model of neuromodulation for the last forty years. Concluding remarks summarizethe potential of these design principles for control. Excitability: a mixed feedback principle
The fundamental dynamical property of neurons is excitability . Excitability of a systemcan be defined as an input-output property by considering the response of the system to pulseinputs of varying magnitude and duration. In the case of neurons, the external applied current istaken as the input variable, while the cell’s membrane voltage is the output. A neuron is excitablebecause its response is all-or-none: for small, subthreshold current pulses, the system exhibits thestable response of a passive circuit with small variations in the membrane voltage. In contrast,a current above a certain threshold causes a spike , or action potential characterized by a large,well-defined excursion of the voltage (Fig. 1). Such specific nonlinear behavior comes from theappropriate balance of ionic currents that flow through the membrane with voltage-dependentdynamics. This mechanism is described in detail in the sidebar ”The Hodgkin-Huxley model”,detailing the generation of the action potential in the squid giant axon.The underlying ionic mechanisms described in the seminal Hodgkin-Huxley paper gener-alize to all excitable neurons, but differ in their structure and complexity. At the core of themodeling of excitability is the representation of a neuron as a nonlinear circuit, known as aconductance-based model.Every neuron is modeled as an electrical circuit describing the dynamical activity of itscellular membrane. This conductance-based structure means that the dynamics of each ioniccurrent can be modeled as a time-varying conductance in series with a battery. The neuralcircuit is then modeled as a parallel interconnection of the membrane capacitor, a passive leakcurrent, and the ionic currents (Fig. 2). Kirchhoff’s current law gives the membrane equation C ˙ V = − I l − (cid:88) j I j = − g l ( V − E l ) (cid:124) (cid:123)(cid:122) (cid:125) Leak current − (cid:88) j g j ( V − E j ) (cid:124) (cid:123)(cid:122) (cid:125) Ionic currents , (1)3here E j is the equilibrium potential for each ionic current j . Each conductance varies between and the maximal value ¯ g j . Inherited from the original model of Hodgkin and Huxley, a commonmodel of the conductance is g j = ¯ g j m pj h qj , (2)where m j the activation variable, h j the inactivation variable, and p and q nonnegative integers.The activation and inactivation variables are confined to the range [0 , , reflecting the openingand closing of the ion channels. They have first-order dynamics: τ m j ( V ) ˙ m j = m j, ∞ ( V ) − m j , (3a) τ h j ( V ) ˙ h j = h j, ∞ ( V ) − h j . (3b)The activation and inactivation steady-state functions are both modeled as sigmoidal functions,but differ in the sign of their derivative: m j, ∞ ( V ) is monotonically increasing, correspondingto activation, while h j, ∞ ( V ) is monotonically decreasing, corresponding to inactivation. Thetime constants are often assumed constant or have a bell-shaped dependence on the voltage(see ”The Hodgkin-Huxley model” for an example). The time constant of inactivation is usuallysignificantly larger than the time constant of activation, as the activation process precedes theinactivation process.Neurons utilize a large bank of ionic currents. The variety of ionic currents provides avariety of control parameters to achieve both flexible homeostatic regulation and robustnessthrough redundancy. This apparent complexity may seem overwhelming. However, all excitablesystems share a remarkably simple and ubiquitous feedback structure. The core question thusboils down to understanding how the simple feedback structure can be extracted from complexneuronal models, and how the effect of each dynamic current maps to the distinct feedbackloops.Two main characteristics of neuron circuits that simplify the analysis are the parallelinterconnection structure of the circuit, which means that ionic currents are additive, and asignificant difference in the speed of activation and inactivation of different ionic currents,allowing for quasi steady-state analysis in distinct timescales. To this extent, standard linearizationtechniques can be utilized. They are described in ”Linearizing conductance-based models”. Thelocal behavior of the circuit around a given voltage is modeled by a parallel circuit interconnectionof linear resistor, inductor and capacitor components, characterized by the total impedance ofthe circuit around the linearization point. The following section illustrates the insight providedby linearization on the Hodgkin-Huxley model, the archetype of excitable neuronal models.4 eedback structure of excitability The classical model of excitability, the Hodgkin-Huxley model, has three feedback currents:a resistive leak current, an inward sodium current, and an outward potassium current. Followingthe linearization procedure, it reduces to the linear circuit shown Fig. 3. The value of eachcomponent depends on the voltage around which the linearization is applied.The interesting insight from this analysis comes by considering the steady state conductancefunctions G ( V ) , g m ( V ) , g h ( V ) and g n ( V ) at every voltage in the physiological range. This isshown in Fig. 4, where the conductances are grouped based on the timescale in which they areacting: • The instantaneous branch describes the passive dissipative property of the membrane. • The fast branch is due to the fast action of the sodium activation, which is several timesfaster than the sodium inactivation and potassium activation. • The slow branch is due to the slower action of the sodium inactivation and potassiumactivation.Linearization illuminates the dynamical signature of excitable systems in terms of thecircuit structure. The conductance in the instantaneous branch is purely positive and models thepassive properties of the membrane (Fig. 4, left). As the sodium current is inward, its activationprovides positive feedback amplification modeled by the negative conductance of the circuit(Fig. 4 middle), while its inactivation conversely provides positive conductance in the slowertimescale (Fig. 4, right). Similarly, potassium is an outward current, so that its activation providesnegative feedback amplification, modelled again as a slow positive conductance (Fig. 4, right).Excitability comes from the interconnection of the passive circuit with a fast negative and a slowpositive conductance.The second important point captured by the linearization is the local action of the dynamiccurrents. Both the fast negative conductance due to g m and the slow positive conductance dueto g h and g n act in a voltage window defined by the their respective gating variables (i.e. wherethe derivatives of m ∞ , h ∞ and n ∞ are non-zero).Analyzing the linearized model therefore highlights the following properties of the currentsof an excitable neuron: • The feedback currents are localized in amplitude : they act in a limited voltage window. • The feedback currents are localized in time : they have a well defined timescale in whichthey act. 5hese two properties allow the currents to shape the circuit’s impedance in a local voltage andtime range. Minimal models of excitability
The fast positive, slow negative feedback structure of excitability poses the question ofa minimal dynamical representation that is sufficient to capture the fundamental behavior ofexcitable systems.The classical minimal model of excitability comes in the form of the FitzHugh-Nagumoequations [27], [28] ˙ V = − V V − n + I app ,τ ˙ n = V + a − bn. (4)The two-dimensional model qualitatively captures the picture of Fig. 4: the instantaneousdissipative properties are captured by the term V , the fast negative conductance by theinstantaneous term V , and the slow positive conductance by the term n . The time constant τ (cid:29) models the timescale separation between the fast variable V and the slow variable n .Model (4) provides a phase portrait of excitability. In addition, the simple structure ofthe model allowed for straightforward circuit realizations, starting with the contribution fromNagumo and colleagues [28].Another important minimal model is the Morris-Lecar model [29]. Unlike the FitzHugh-Nagumo model, the model has a conductance-based structure which is why it has commonlybeen used when biophysical interpretability is needed. The model is minimal as it is two-dimensional and has the necessary and sufficient components for excitability: a fast negativeconductance component, represented by the instantaneously activated inward calcium current, andslow positive conductance component, represented by the slowly activating potassium current.The model is described by C dVdt = − g l ( V − E l ) − g Ca m ∞ ( V )( V − E Ca ) − g K n ( V − E K ) + I app ,τ n ˙ n = n ∞ ( V ) − n. (5)From the dynamical systems perspective, the model is also important because for differentparameters it can model different types of excitability in a minimal representation amenableto phase plane analysis.The phase portraits of the two models are discussed in ”I-V curves and phase portraits”.6 ultiscale excitability: burst excitability The fundamental excitability motif of fast positive, slow negative feedback may appearrepeated in a slower timescale, endowing the neuron with slow excitability in addition to theexcitable properties discussed previously. Slow excitability was discovered experimentally soonafter the work of Hodgkin and Huxley [30]. It plays an important role in the regulation ofexcitability. This multiscale excitability motif leads to the specific form of excitable behaviorcalled bursting . Bursting is a prevalent signaling mechanism in neurons [12], [31]–[35] andhas important consequences on their synchronization properties as well as their informationprocessing capabilities.From the input-output perspective, burst excitability manifests as an all-or-none responsein the form of a burst of spikes. This characteristic behavior is shown in Fig. 5. Like in thecase of single-scale excitability, there is no unique physiological model of a bursting cell, anddifferent neurons can have vastly different combinations of ionic currents that generate theiractivity. Still, an elementary feedback structure is common to all of them. A burst output inresponse to an input pulse consists of a slow spike that in turn activates fast spiking activityduring the upstroke phase of the slow response. Thus, the underlying dynamical structure ofbursting consists of a parallel interconnection of fast excitable and slow excitable subsystems,orchestrated by at least four distinct ionic currents. Linearization of conductance-based modelsof bursting neurons illuminates this dynamic structure as shown in the following section.
Feedback structure of burst excitability
The feedback structure of bursting neurons is illustrated on the neuron R-15 of Aplysia[36], [37]. The model has been one of the most extensively studied bursting neurons in theliterature. The neuron has the same ionic currents as the Hodgkin-Huxley model, i.e. the sodiumand potassium channels responsible for generating individual spikes, but in addition it has slowerchannels that are responsible for generating the slow variations of the membrane voltage. Thesechannels are responsible for two additional ionic currents: an inward calcium current, and anoutward calcium-activated potassium current. The model is described by
C dVdt = − I l − I Na − I K − I Ca − I K − Ca + I app , (6)where I l , I Na , and I K are as in (S3) with the standard simplification that sodium activation isinstantaneous (i.e. m = m ∞ ( V ) ). The additional currents are modeled as g Ca = ¯ g Ca x, (7) g K − Ca = ¯ g K − Ca c . c . (8)7he calcium activation variable x and the calcium concentration c can be written in the standardform of (3). Note that the calcium-activated potassium current does not have standard form of(2) but it can be analyzed in the same way through linearization.The time constants of the additional currents in the model τ x and τ c are significantly largerthan the dynamics of the Hodgkin-Huxley currents. The calcium current I Ca therefore activatesslowly compared to the potassium current I K , while in turn the calcium-activated potassiumcurrent I K − Ca is the slowest in the model.The two additional currents of the bursting model are modeled differently than in theHodgkin-Huxley model but share the exact same mixed feedback structure to generate slowexcitability . The calcium current is is a slow analog of the sodium current: it is inward and itsactivation leads to a positive feedback loop. Likewise, the calcium-activated potassium currentis a slow analog of the potassium current: it is outward and its activation provides negativefeedback in the slowest timescale of the model. Bursting can therefore be effectively regardedas duplicating the feedback structure of Hodgkin-Huxley model: fast spiking results from thecombination of fast positive and slow negative feedback, while bursting requires the same mixedfeedback structure, but in a slower timescale. This structure is reminiscent of outer and innercontrol loops encountered in many feedback control systems.The structure of the linearized circuit is illustrated in Fig. 6. Adding to the picture shownin Fig. 3, the two currents introduce two resistor-inductor branches to the linearized equationand their voltage dependence can be plotted as previously (Fig. 7).Again, linearization illuminates the basic circuit structure of a bursting neuron. The fastnegative , slow positive conductance of excitability is effectively repeated twice, so that the fastercombination of the conductances generates the individual spikes within the burst, while theslower conductance combination generates the slow wave that switches the spiking on and off.The nested structure is also reflected in the different localization in the voltage range: thefast conductances are localized in a higher voltage range than the slower conductances, whichenables the distinct activation of the slow and fast thresholds. Minimal models of burst excitability
There are various proposed mathematical models of bursting in literature, capturing theparticular dynamical features of different neuronal bursting cells [38]. From the mathematicalviewpoint, the main classification between bursting types lies in the mechanism of the transitionfrom resting to spiking and back. If the fast spiking system has a region of bistability betweenthe resting and spiking states, then an additional slower variable is sufficient to drive the system8round the hysteresis and generate a bursting attractor. However, if the fast system is monostable,then the slow system needs to be at least two-dimensional in order to generate semi-autonomousslow oscillations that push the system between spiking and rest. Analysis of these mechanismshas led to an extensive classification that focuses on the bifurcations that lead to the transitionbetween resting and spiking and vice-versa [39].An early and key example of a minimal bursting model came from the analysis ofHindmarsh and Rose [40]. The model can be seen as a modification of the minimal spiking modelof FitzHugh that establishes bistability between stable rest and spiking states by introducing aquadratic slow variable instead of a linear one. The variable thus captures both negative andpositive feedback in the slow timescale, combining the two slow conductances of Fig. 7 in a singlevariable. In addition to modifying the slow variable, the model introduces ultra-slow adaptationthat provides negative feedback in the ultra-slow timescale and generates the oscillation betweenthe rest and spiking states. The full model is ˙ V = − aV + bV − n − z + I app , ˙ n = dV − c − n,τ z ˙ z = s ( V − v ) − z. (9)Recent work has illuminated the important fact that regardless of the underlying mechanismof bursting, in order to robustly model the dynamical features of bursting neurons, it is necessaryto retain the essential four feedback loops discussed in the previous section [26], [41]–[43]. Thishas led to the proposed three-timescale model based on the transcritical singularity [23]. Themodel is based on the reduction of the Hodgkin-Huxley model with the addition of a slowly-activated calcium current [20], and it shares a similar structure as the FitzHugh-Nagumo modelof spiking and the Hindmarsh-Rose model of bursting. In addition to these well-known analysis,the model provides a mathematical framework for understanding the mechanisms that allow asystem to change its behavior between bursting and spiking modes. The model is given by ˙ V = − V V − ( n + n ) − z + I app ,τ n ˙ n = n ∞ ( V ) − n,τ z ˙ z = z ∞ ( V ) − z. (10)Both model (9) and model (10) can be studied by treating the ultra-slow variable z asa constant parameter and analyzing the family of phase portraits parameterized by z . Burstingis obtained from slow adaptation of z over the range of parameters where the phase portraitsexhibit bistability between spiking and resting. See ”I-V curves and phase portraits” for details.9 euromodulation as a control problem Neuromodulation is a general term describing the effects of various neurochemicals suchas neurotransmitters and neuromodulators on the excitability of neurons. Neuromodulators haveprofound effects on the behavior of neural networks [13], [14], [31], [44], and understandingthe effects of neuromodulation from a control perspective is essential in order to characterizethe coexisting robustness and sensitivity of neuronal behaviors.The detailed molecular mechanisms of neuromodulation are complex, but their aggregateeffect at the cellular scale is that they modulate both the expression of ion channels as wellas the sensitivity of their receptors to various neurotransmitters. In the electrical circuit modelof a neuron, the effect of neuromodulation is modeled by varying the maximal conductanceparameters of the currents ¯ g j . Understanding how neuromodulators can drive the behavior inthe desired way therefore boils down to understanding which regions in the parameter spaceof the maximal conductances correspond to which behavior. This is a difficult task due to thehighly nonlinear nature of the currents, but recent work [25] has proposed a way of viewingthe modulation of maximal conductances as a form of loop-shaping of the basic feedback loopsresponsible for generating the behavior.The sensitivity of most parameters to a specific neuromodulator can be superimposedthrough the linearization technique into the aggregate conductances for each timescale: fast , slow and ultra-slow . Spiking then requires the appearance of fast negative conductance, and slowpositive conductance, while bursting requires the addition of the slow negative conductance andultra-slow positive conductance, appropriately localized in the voltage range.Modulating the maximal conductances thus shapes the conductance in each timescale,changing the properties such as frequency, duty cycle, as well as the qualitative properties of thewaveform. The critical properties of both fast and slow excitability are primarily controlled bybalancing the total positive and negative conductances in the fast and the slow timescale, whichprovides a highly redundant tuning mechanism, robust to large uncertainties in the individualelements.For any proposed neural model, its modulation can be viewed as shaping the balance ofpositive and negative conductances in the appropriate timescale. This viewpoint is applied tothe neuromorphic design in the following sections by proposing that it is both necessary andsufficient to appropriately model the conductance in the three relevant timescales.10 ynthesis and control of neuromorphic circuits The neuromorphic approach aims to understand and mimic the biophysical mechanismsunderlying neuronal behavior in order to build electrical systems with similar capabilities. Theinitial interest in bioinspired systems has sparked a rapidly developing research field (see ”Threedecades of neuromorphic engineering”). Neuromorphic hardware provides an exciting venuefor bioinspired control systems, see for instance the recent survey [2]. In spite of some earlyexamples of joint efforts between the two communities (”The first neuromorphic controller”),it is fair to say that the potential for novel biologically motivated control methods still remainslargely unexplored.The aim of this article is to simplify the apparent complexity of biophysical models throughthe prism of the feedback loops essential to excitability. This section presents a neuromorphicdesign and control methodology inspired by the biophysical conductance-based structure ofneuronal models. It shows how the fundamental dynamical features of spiking and burstingneurons can be recreated in a circuit structure which is simple to manufacture and controlwith existing neuromorphic components. At the core of the framework are elementary feedbackcurrent elements that provide local positive or negative feedback, in a similar vein to the roleof individual ionic currents in physiological models of neurons. The model provides a way ofsynthesizing electrical circuits with the mixed feedback structure that can be analyzed by shapingits input-output characteristic in the form of I-V curves.The I-V curve of a circuit is an input-output characteristic, independent of its state-space realization. The traditional I-V curve of a circuit is its static characteristic, providingthe constant operating points of the circuit. It is a common experimental characterization ofneurons in electrophysiology. Instead, we consider distinct I-V curves in distinct timescales andwe show that this representation as a close link to the classical phase portrait analysis of neuronalbehaviors (see ”I-V curves and phase portraits”). In that sense, the framework of this sectionpresents an introduction to synthesizing neural behaviors in hardware by connecting the simpleinterpretability of the classical phase portrait models and the physiological membrane models. Ituses the simplified neuromorphic architecture to show the main input-output conditions that needto be satisfied in excitable neuronal models, and discusses how the technique can be extendedto more general models of the excitable neural membrane.
Neuromorphic architecture
Mirroring the neuronal membrane organization where the ionic currents form a parallelinterconnection structure, the proposed neuromorphic architecture also consists of a capacitorinterconnected in parallel with a single resistive element, and a set of circuit elements emulating11he active ionic currents. The general representation of the circuit is shown in Fig. 8.The resistive element has a monotonically increasing input-output relationship dI p ( V ) dV > , for all V. (11)Each element I j provides positive or negative conductance in a given voltage range and a giventimescale. Each element is modeled as the series interconnection of a linear filter and a nonlinearactivation function multiplied by the maximal conductance parameter: I j = g j ( V x )( V − E j ) , (12a) g j ( V x ) = ¯ g j S j ( V x ) , (12b) τ x ˙ V x = V − V x . (12c)Here, the current is a function of the filtered voltage V x which can be common to multipleelements and S j is a sigmoidal activation function.The model (12) is a simplified conductance-based model that has been considered inrecent work [11]. For the purpose of introducing input-output analysis techniques, a furthersimplification is applied to the physiological models by discarding the ohmic form of the currentsand modeling every element as a current source whose output depends solely on a time-filteredversion of the membrane voltage. This means that elements have no instantaneous dependenceon the membrane voltage V . Thus, each element has the input-output characteristic I j = f j ( V x ) , (13a) τ x ˙ V x = V − V x . (13b)This is the model that is the focus of the following sections as it allows for a simplified input-output analysis described in the following section. The difference between (12) and (13) isquantitative rather than qualitative and a design based on the simple model (13) is easily translatedinto a conductance-based implementation (12).A convenient way to model the input-output functions f j is to separate the elements intopurely positive conductance elements I + j and purely negative conductance elements I − j thatwould therefore have the input-output characteristics df + j ( V ) dV > , for all V,df − j ( V ) dV < , for all V. (14)In addition, the elements can be grouped to act solely in a few specified timescales. Thisenables the analysis of the circuit through well-known techniques that utilize the timescale12eparation of the subsystems [45]. In the previous section it has been highlighted that neuronalspiking is fundamentally a two-timescale phenomenon, while neuronal bursting is fundamentallya three-timescale phenomenon. By introducing the timescale of the membrane equation C/τ V = dI p ( V ) dV (cid:12)(cid:12)(cid:12)(cid:12) V = V rest , (15)the elements can be grouped into three separate timescales: fast ( τ f ), slow ( τ s ) and ultra-slow ( τ us ) satisfying the condition max( τ V , τ f ) (cid:28) τ s (cid:28) τ us . (16)The fast dynamics is usually taken to be of similar order as the dynamics of the voltage equation,and a commonly applied simplification is for the fast element dynamics to be instantaneous, i.e. τ f = 0 . This assumption is used in the synthesis of excitable neuronal circuits in the followingsections. In this setting every element can be indexed by the timescale in which it acts and thesign of its conductance, so that I j = I ± x ∈{ f,s,us } . (17)For a general neuromorphic model it is convenient to use the same form for all currents I j , as well as consider an input-output mapping that is easily realizable in hardware, which isnot necessarily the case for general polynomial forms. In this regard, a suitable choice for thefunctions f − j and f + j is a hyperbolic tangent function, so that f + j = α j tanh( V x − δ j ) ,f − j = − α j tanh( V x − δ j ) . (18)The function is defined by two parameters α j and δ j and is convenient for several reasons.Due to the saturation effect, the effect of each current is localized in the voltage range withthe parameter δ j controlling the position of the voltage window in which it acts, while theparameter α j controls the strength of the current. The localization property is important as itcaptures the localized scope of the activation and inactivation functions. Secondly, hyperbolictangent I-V characteristic is easily realizable in hardware, as presented in ”Using physics tocompute: subtreshold electronics” and ”Neuromorphic building blocks”. I-V curve characterization
The structure of the model (13) converts the analysis of the circuit into the shaping ofits input-output characteristic or so-called I-V curves. In addition to the static I-V curve whichis measured by fixing the input voltage and measuring the corresponding static output current,transient properties can be characterized by defining the I-V curves in several distinct timescales.For this purpose, analysis can be conducted in the three timescales that are sufficient to capture13piking and bursting properties of neurons: fast, slow and ultra-slow. The corresponding fast,slow and ultra-slow I-V curves are then defined as I x ( V ) = I p ( V ) + (cid:88) τ y ≤ τ x f ± y ( V ) , (19)so that I x sums the input-output characteristics of all currents acting on the timescale τ x orfaster.This set of input-output curves is sufficient for the analysis of the circuit due to the clearseparation of the timescales in the model: for the quasi steady-state analysis of the model intimescale τ x it is possible to freeze the effects of the currents with slower dynamics. Thissimplification enables the synthesis of excitable circuits to reduce to simple graphical conditionson its input-output I-V curves. In particular, generating excitable behavior involves introducing aregion of negative conductance in a given timescale and restoring the positive conductancein a slower timescale, achieved by the appropriate interconnection of positive and negativeconductance feedback elements. Controlling the circuit’s behavior thus involves shaping thenegative conductance region in different voltage ranges and timescales, enabling the control ofmultiscale excitable behaviors.For general conductance-based models it is not possible to define the I-V curves inthe simple sense of (19), as the clear division between the timescales of the model is notalways achievable and each current may depend on multiple gating variables, as well as theinstantaneous membrane voltage. Still, the intuition from the synthesis and control of thesimplified neuromorphic model can be applied to the more complex physiological models throughappropriate reductions [25], [46]. The procedure involves setting the representative timescales,and grouping the effects of ionic currents into weighted sums for fast ( g f ), slow ( g s ) and ultra-slow ( g us ) differential conductances. The reader will note the relationship between the I-V curvesand the feedback gains of the linearized analysis: the slope of the I-V curve in a given timescaleis the sum of the linearized conductances in the same timescale and all faster timescales. Thismeans that shaping the I-V curves of the neuromorphic circuit is equivalent to shaping thelinearized conductances, that is, the voltage-dependent feedback gains of the circuit in eachtimescale. Excitable circuit
In ”Feedback structure of excitability” the core feedback components of an excitableneuronal membrane were highlighted to be an ionic current acting as a source of fast positivefeedback, and a slower ionic current providing negative feedback. The minimal neuromorphiccircuit is shown in Fig. 9. 14he fast element I − f generates a region of negative conductance in the fast I-V curvedefining a region of bistability in the fast timescale (Fig. 10). The slow element I + s restores themonotonic I-V characteristic in the slow timescale by balancing the negative conductance of thefast I-V curve with positive conductance characteristic. The sufficient conditions for generatingexcitable behavior are thus d I f dV < , V ∈ ( V f , V f ) , (20) d I s dV > , for all V, (21)which are shown in Fig. 11.Given that the separation of timescales is satisfied (16), the resulting behavior has thefollowing characteristics: • The steady-state characteristic is given by the monotonic slow I-V curve, so that for eachconstant input I app there is a unique equilibrium point ( I app , V e ) . The equilibrium is stableexcept for a finite range of voltages within the interval ( V f , V f ) . • The circuit has a stable spiking behavior for a finite range of constant applied input I app .The spiking behavior is characterized by stable oscillations of the relaxation type, with theamplitude of the oscillation determined by the bistable range of the fast I-V curve (Fig. 10). • For stable equilibria near the unstable region contained in ( V f , V f ) , the system is in the excitable state: short input pulses can trigger individual output spikes , i.e. the transientmanifestations of the spiking behavior.The circuit in Fig. 9 shares the same essential structure as the FitzHugh-Nagumo modelwhere the oscillatory spiking behavior appears through a subcritical Hopf bifurcation nearthe turning points of the fast I-V curve V f and V f . The unstable range converges to therange ( V f , V f ) as the timescales are more sharply separated, i.e. as max( τ V , τ f ) /τ s → . Themonotonicity of the slow I-V curve in conjunction with the N-shaped fast I-V curve and thesufficient separation in timescales guarantees the onset of spiking to appear through a Hopfbifurcation.The specific bifurcation mechanism by which the rest-spike transition happens has impor-tant information processing consequences for the neuronal behavior [46]. When the transitionhappens through a Hopf bifurcation, the oscillations emerge with a non-zero frequency. Instead,some neurons have a continuous frequency curve as the input is varied. This property is attributedto oscillations appearing through a saddle-node bifurcation [38], requiring non-monotonicity ofthe slow I-V curve. By ensuring that the turning point of the fast I-V curve coincides withthe turning point of the slow I-V curve, the transition between rest and spiking instead appears15hrough a saddle-node on an invariant circle bifurcation (SNIC). Figure 12 shows a comparisonof the frequency curves for the two mechanisms.The I-V curve interpretation is directly related to the standard phase portrait analysis asdiscussed in ”I-V curves and phase portraits”. Bursting circuit
Mirroring the structure of bursting neurons (see ”Feedback structure of burst excitability”),a minimal neuromorphic bursting circuit is constructed by the interconnection of the excitablecircuit discussed previuosly (Fig. 9) with the additional feedback elements generating slowexcitability. This structure is shown in Fig. 13.Slow excitability is generated with the same excitability interconnection structure of apositive feedback loop balanced by a slower negative feedback loop. Unlike fast excitability,slow excitability consists of the positive feedback acting on the slow timescale instead of beinginstantaneous, and the negative feedback is ultra-slow.Similarly to how I − f is responsible for generating bistability between a ”low” and a ”high”voltage state in the fast timescale through the N-shaped fast I-V curve, I − s introduces a regionof negative conductance in the slow I-V curve that gives rise to the bistability between a ”low”voltage state and the ”high” spiking state, i.e. rest-spike bistability. This is shown in Fig. 14.Both curves have regions of negative conductance, and importantly, the slow threshold is belowthe fast threshold, i.e V s < V f . (22)Burst excitability is then achieved through the addition of the ultra-slow positive conduc-tance element, and similarly to the case of an excitable system from previous section, a sufficientcondition is for it to restore monotonicity in the ultra-slow I-V curve. The characteristics of aneuromorphic bursting circuit are shown in Fig. 15.Given that the separation of timescales is again satisfied (16), the resulting behavior hasthe following characteristics: • The steady-state characteristic is given by the monotonic ultra-slow I-V curve, so that foreach constant input I app there is a unique equilibrium point. The equilibrium is stable exceptfor a finite range of voltages within the interval ( V s , V f ) . • The circuit has a stable limit cycle characterized by the alternation between quiescence andspiking (bursting) for equilibrium voltages within the region ( V s , V s ) , and a stable spikinglimit cycle within the range ( V s , V f ) . 16 For stable equilibria near the turning point V s the system is in the burst excitable state:short input pulses can trigger a burst of spikes. i.e. the transient manifestation of the burstingbehavior.The dynamical properties of the model are grounded in the analysis of a similar state-spacemodel presented in [20], [21], [23]. The connection with the phase portrait analysis of this modelis discussed in ”I-V curves and phase portraits”. Control of neuronal properties
Each circuit element of the model (13) has a simple interpretation: it provides eitherpositive or negative feedback in a specific voltage range and a specific timescale. The naturalcontrol parameters of this architecture are the maximal conductance parameters. Figures 16and 17 illustrate that modulating those few parameters indeed shapes the spiking and burstingproperties of the circuit. The available applet [47] allows the interested reader to explore the easeof tuning the circuit properties by modulating the maximal conductance parameters. This tuningmethodology is neuromorphic in that the biophysical mechanism of neuromodulation preciselymodulates the maximal conductances of specific ionic currents.A parameter of particular importance is the balance between positive and negativeconductance in the slow timescale. It is indeed the range of slow negative conductancewhich controls the transition between slow spiking (characterized by the absence of negativeconductance in the slow I-V curve) and bursting (characterized by the presence of negativeconductance in the slow I-V curve).The transition between slow spiking and bursting will be shown to be of particularsignificance in the following sections. Modeling the transition between spiking and burstinghas been a bottleneck of many mathematical models of bursting. It is for instance presentedas an open question on the Scholarpedia webpage on bursting [48]. The reason is that manymathematical models of bursting lack a source of negative conductance in the slow timescale.The reader will note that there is no possibility of tuning slow negative conductance without thecurrent source I − s . In contrast, the current source I − s directly controls the transition.The point of transition between the two states can be studied mathematically by consideringthe local analysis of the slow I-V curve around its critical points. This interpretation can bedirectly derived from the singularity theory analysis of the phase plane model in [18], [23]. Byaligning the threshold points of the fast and the slow I-V curves V f = V s , (23)the transition between the two regimes reduces to a condition on the concavity of the slow I-V17urve around this point: in the bursting regime, the curve is locally concave, and in the spikingregime, the curve is locally convex. The two are separated by the condition d I s dV = 0 , (24)which corresponds to the pitchfork bifurcation in the analysis of the phase plane model [23].This is illustrated in Fig. 18 which shows the control of the neuronal behavior through the soleparameter α − s controlling the gain of the slow negative conductance element.The circuit structure of Fig. 13 can be easily realized in hardware using the basic principlesof neuromorphic engineering, and the sidebar ”A neuromorphic neuron in hardware” shows howstandard circuit components can be utilized to implement a circuit implementation amendableto the neuromodulation principles discussed in this section. Why control the excitability of a neuron?
The previous sections have focused on controlling the excitability of a single neuron. Butwhy should we devote so much attention to the details of a single neuron, given that neuronsare elementary units of large networks? The second part of this article addresses this question.Our aim is to show that controlling the excitability of a neuron is a powerful nodal controlmechanism in a network setting.Controlling the excitability of a neuron drastically alters its response to external stimuli[12], [33]–[35]. This not only changes the input-output processing characteristics at the single-neuron level, but changes the synchronization susceptibility of the individual nodes in the networksetting. This excitability switch offers a fundamental mechanism by which individual nodes canlocally control the global network behavior through the appropriate mixed feedback balance atthe cellular level.The section first addresses how neurons are interconnected to form networks, mirroringthe modeling framework of the single neuron feedback elements in ”Synthesis and control ofneuromorphic circuits”. The important input-output mechanisms at the single-neuron level thatare essential for the network control are then highlighted and discussed. The role of thoseinput-output properties will then be illustrated in a two-neuron inhibitory circuit, an elementaryoscillatory module that is a ubiquitous building block of more complex network structures.
Neural interconnections
Even though single neurons exhibit complex behavior on their own, they never function inisolation. Instead, they are a part of richly interconnected neural networks where the behavior18s critically dependent on the dynamical properties of the connecting elements.Similarly to the feedback elements on the single-neuron level, neural interconnectionsconsist of passive interconnections as well as active excitatory and inhibitory interconnections.Both types of connections are prevalent in biology with distinct effects on the net activity: theresistive connections are dissipative and tend to average out the activity across neurons, whilesynaptic connections can lead to diverse heterogeneous behaviors prevalent in neural activity.Passive interconnections are simply modeled as resistive connections between the voltagenodes of neurons, so that the currents flowing into neuron j ( I pij ) and neuron i ( I pji ) due to apassive connection between them are: I pij = − I pji = g pij ( V i − V j ) , (25)where g pij is the conductance of the passive connection. They are the network analog of the leakycurrent of isolated neurons.The active connections model the effects of chemical synapses which act as directedconnections where the activity of the presynaptic neuron affects the behavior of the postsynaptic neuron. They are the network analog of the active current sources of an isolated neuron. Suchcomponents can be modeled in the same framework of (18) as current source elements in parallelwith the membrane elements of the postsynaptic neuron whose output depends on the voltageof the presynaptic neuron. The synaptic current I synij in neuron j thus depends on the activity ofneuron i as I synij = α synij S ( β ( V xi − δ syn )) , (26a) τ x ˙ V ix = V i − V xi , (26b)where S ( x ) is the sigmoid function, α synij is the synaptic weight, β is the steepness factor, δ syn is the voltage offset of the synaptic current. This form is used to model both inhibitory andexcitatory synaptic connections with the only difference lying in the sign of the synaptic weight α ij . In the case of an excitatory synapse, the spike in the presynaptic neuron induces an inwardcurrent I excij in the postsynaptic neuron so that α ij > , (27)and in the case of an inhibitory synapse, the spike in the presynaptic neuron induces an outwardcurrent I inhij in the postsynaptic neuron, so that α ij < . (28)The reader will note that the only difference between intrinsic and extrinsic current sourcesis that the current depends on the intrinsic voltage in the former case, while it depends on the19xtrinsic voltage in the latter case. The emphasis in the design of artificial neural networks isusually on controlling the extrinsic (or synaptic) conductances. The aim of the present articleis to highlight the equally important role of intrinsic conductances for the control of networks.Biophysical neural networks rely as much on the modulation of intrinsic nodal feedback loopsas on the modulation of the network synaptic connectivity [44]. Controlling the relay properties of a neuron: a nodal mode and a network mode
The robustness of neural communication lies in the excitable nature of neurons. Thethreshold property of firing neurons allows neural connections to reliably send information aboutsignificant events across long distances in spite of the signal degradation and noise characteristicof analog systems. Controlling the excitability of a neuron modulates the properties of this relaytransmission.The combination of the fast and slow excitability feedback elements endows the neuronwith four possible discrete dynamical states, depending if the slow and fast excitability arepresent. Of particular importance is the control of the slow excitability of the node while thecell remains fast excitable. This mechanism manifests as the transition between bursting and slowspiking behaviors presented in Fig. 18. In the first approximation, the role of fast excitability canbe interpreted as to relay action potentials between neurons, whereas the role of slow excitabilityis to modulate the gain of the relay transmission. This is why the transition between slow spikingand bursting was emphasized in the single neuron analysis.Figure 19 illustrates the significance of this transition for the relay properties of a neuron.In the slow spiking state, the system acts as a relay to the individual spiking events: input spikesdirectly correlate with the spikes at the output. In contrast, in the bursting mode, individualspikes are effectively low-pass filtered and do not appear at the output.This excitability switch has important consequences for the single cell behavior in thenetwork setting. In the bursting mode, a neuron tends to synchronize with the slow rhythm ofthe network and effectively blocks the transmission of local spiking inputs. In contrast, in thespiking mode, a neuron can switch off from the network rhythm and participate in the localrelay of spiking inputs. This elementary mechanism suggests a key role for the slow negativeconductance of a neuron: it controls the transition between a nodal mode, where the neuronrelays local information at the cellular scale, and a network mode, where the individual neuronbecomes an internal unit of a larger ensemble.20 ontrolling the rebound properties of a neuron
In its nodal mode, the input-output response of a neuron is primarily characterized byFig. 1, that is, by how it responds to a short depolarizing input current. In contrast, how aneuron responds to a longer hyperpolarizing pulse turns out to be critical for its network mode.This property is known as the post-inhibitory rebound . A robust generation of the post-inhibitoryrebound response is essential for the generation of oscillations in networks where the individualneurons do not oscillate in isolation (see ”Central pattern generators”).In neurophysiology this mechanism has been linked to two specific inward currents,known as the ’hyperpolarization-activated cation current’ I h and the ’low-threshold T-typecalcium current’ I Ca,T [49]. The low threshold inward calcium current only activates underhyperpolarization. Its activation provides the slow negative conductance necessary for bursting.The I h current adds to the total inward current to regulate the response to hyperpolarization.The rebound mechanism is first illustrated in Fig. 20 on a spiking neuron. Starting fromthe bistable circuit of Fig. 10, the rebound spike is obtained by adding a slow negative feedbackcurrent with low activation range. In the same manner, the rest-spike bistable system of Fig. 14can provide a transient bursting response with an additional ultra-slow negative feedback currentFig. 21. Turning a rhythmic circuit on and off
The post-inhibitory rebound input-output response of a neuron was recognized more than acentury ago as central to rhythm generation in neuronal circuits (see ”Central pattern generators”).The simplest illustration is provided by the circuit made of two neurons that inhibit each other.The anti-phase rhythm illustrated in Fig. 22 is a direct consequence of the rebound properties ofeach neuron. Even if the two neurons do not burst in isolation, their mutual inhibition generatesan autonomous rhythm in the circuit. Each neuron of this network exhibits its network mode,that is, becomes an internal component of a rhythm at the circuit scale.The two inhibitory connections define the phase relationship between the individualoscillations so that the neurons synchronize in anti-phase, even when the neurons oscillateendogenously. The excitability switch shown in Fig. 19 effectively switches the neuron froman endogenous source of oscillation, robust to external disturbance, to an exogenous relay mode,sensitive to external inputs.In Fig. 23, the gain of the slow negative conductance of the individual neurons is high andthey generate robust synchronized bursting oscillations. When subjected to external disturbancein the form of train of pulses in the applied currents, the behavior is largely independent of the21hanges at the inputs. Lowering the gain of the slow negative conductance of the two neurons(Fig. 24), the endogenous bursting oscillation is turned off, and the neurons act as independentrelaying systems. The effect of the input train of pulses is drastically different in this mode, asindividual spikes are all transmitted.The nodal excitability switch provides a way of controlling the network rhythm through acontrol of a single nodal parameter. By scaling up the same principles, more complex networksthat consist of interconnections of such rhythm-generating modules can switch between differentrhythms through the control of excitability properties of the individual nodes.The reader will note that the switch between the nodal and network mode of a neuron isitself a control mechanism that can be intrinsic or extrinsic to the neuron: it is intrinsic whencontrolled via the modulation of the slow negative conductance of the neuron, and is extrinsicwhen controlled via hyperpolarization of the neuron through the applied current. The versatilecontrol of this neuronal switch by a diversity of different mechanisms indeed suggests its keysignificance in controlling the properties of a neural network.
Nodal control of a network
This final section aims at illustrating that the basic control principle described in theprevious section is eminently scalable and modular. A network can be dynamically reconfiguredinto a variety of different networks by simply modulating which nodes are ”in” at a giventime and a given location. Here, a five-neuron network is considered, inspired by the well-known crustacean stomatogastric ganglion (STG) structure [50]–[52]. The STG has been a centralbiological model over the last forty years in the discovery of the significance of neuromodulation.The interested reader is refered to the inspiring essay [53], a fascinating account of the scientificjourney of Eve Marder into the world of neuromodulation as a key control principle of biologicalnetworks. The network below is a crude simplification of the biological STG network, but retainsits core structure: it consists of an interconnection of two central pattern generating circuits thatoscillate at different frequencies. The central pattern generating networks are modeled by twotwo-neuron inhibitory modules that oscillate in anti-phase. The two CPGs interact with eachother through resistive connections with a middle ”hub” neuron, and thus the network rhythmis given by a combination of the two individual rhythms.The network is recreated with the neuromorphic model described in ”Synthesis and controlof neuromorphic circuits” in Fig. 25. The parameters of the neurons are chosen so that the twoinhibitory circuits generate a fast and a slow rhythm, respectively. The difference between thetwo subnetworks involves no change in the time constants of the individual currents, but only22 different value of the ultra-slow feedback gain, which was shown to control the interburstfrequency in the single neuron analysis. The middle hub neuron is set to its slow spiking mode,so that it acts as a relay between the two independent rhythms.Figure 26 shows the two rhythms interacting under the addition of resistive connectionsbetween the hub neuron and the individual CPGs. When the resistive connections are weak, theindividual rhythms are mostly unaffected by the connection, and the fast and the slow rhythmwithin the network coexist. The hub neuron exhibits a mix of the two frequencies.By using the mechanism shown previously in Fig. 24, the individual rhythms can beeffectively disconnected from the network by modulating the two-neuron oscillators between thebursting and slow spiking modes. This is shown in Fig. 27 where the slow negative conductancegain is decreased for the neurons within the slow oscillator. The slow oscillator neurons then actas followers, and the fast rhythm propagating through the hub neuron now has sufficient strengthto synchronize these neurons. In this case the whole network oscillates with the fast rhythm.A more extensive investigation of the various configurations of the circuit is providedin [43]. The key message is that the control of each node in its nodal versus network modesenables dynamic transitions between networks that share the same hardware network topologybut exhibit a different functional topology controlled by which node is ”in” (network mode) andwhich node is ”out” (nodal mode). This suggests that the control of excitability at the nodallevel provides a rich network control principle.
Conclusion
The article presented a framework for designing and controlling mixed-feedback systemsinspired by biological neuronal networks. This neuromorphic approach aims at mimicking thefeedback structure, as well as the control mechanisms that make neurons robust, adaptable andenergy efficient.The dynamical structure of excitable neurons was presented through the fundamentalfeedback organization consisting of interlocked positive and negative feedback loops acting indistinct timescales. By using well-known low-power analog circuits, these elementary feedbackloops can be efficiently implemented in hardware while retaining the interconnection structureof biological neurons. At the same time, their unique dynamical structure makes neuromorphicneurons amendable to an input-output analysis that reframes the control problem as static loopshaping of the input-output characteristic in distinct timescales.The article further discussed how this mixed-feedback organization of the neuronal nodes23ndows them with a unique dynamical switch in the network setting. Biological neural networksconstantly adapt to the changing conditions and requirements through appropriate modulationof neurons and their interconnections. By controlling the excitability of the nodes, the articleshowed how individual circuits can turn local rhythms within the network on and off, offeringa robust mechanism for controlling the global network behavior through the appropriate mixedfeedback balance at the individual nodes.The main obstacle in the development of neuromorphic circuits has proven to be theirincreased fragility to noise and fabrication deviations that is an inevitable consequence of theiranalog mode of operation. However, biological neurons effectively cope with the same limitationsthrough their unique multiscale mixed-feedback architecture. By combining the expertise of bothcontrol and electronics experts, neuromorphic control shows great promise in helping to solvefuture large-scale challenges in energy efficient ways using direct inspiration from biology.24 eferences [1] C. Mead, “Neuromorphic electronic systems,”
Proceedings of the IEEE , vol. 78, no. 10,pp. 1629–1636, 1990.[2] G. Gallego, T. Delbruck, G. Orchard, C. Bartolozzi, B. Taba, A. Censi, S. Leutenegger,A. Davison, J. Conradt, K. Daniilidis, and D. Scaramuzza, “Event-based Vision: A Survey,”
IEEE Transactions on Pattern Analysis and Machine Intelligence , pp. 1–1, 2020.[3] C. Mead, “How we created neuromorphic engineering,”
Nature Electronics , vol. 3, pp. 434–435, July 2020.[4] G. Indiveri, B. Linares-Barranco, T. J. Hamilton, A. van Schaik, R. Etienne-Cummings,T. Delbruck, S.-C. Liu, P. Dudek, P. H¨afliger, S. Renaud, J. Schemmel, G. Cauwenberghs,J. Arthur, K. Hynna, F. Folowosele, S. Saighi, T. Serrano-Gotarredona, J. Wijekoon,Y. Wang, and K. Boahen, “Neuromorphic Silicon Neuron Circuits,”
Frontiers in Neuro-science , vol. 5, 2011.[5] G. Cauwenberghs, “Reverse engineering the cognitive brain,”
Proceedings of the NationalAcademy of Sciences , vol. 110, pp. 15512–15513, Sept. 2013.[6] S. Furber, “Large-scale neuromorphic computing systems,”
Journal of Neural Engineering ,vol. 13, p. 051001, Oct. 2016.[7] K. Boahen, “A Neuromorph’s Prospectus,”
Computing in Science & Engineering , vol. 19,no. 2, pp. 14–28, 2017.[8] C. Mead,
Analog VLSI and Neural Systems , vol. 1. Reading: Addison-Wesley, 1989.[9] R. Sarpeshkar, “Analog Versus Digital: Extrapolating from Electronics to Neurobiology,”
Neural Computation , vol. 10, pp. 1601–1638, Oct. 1998.[10] C.-S. Poon and K. Zhou, “Neuromorphic Silicon Neurons and Large-Scale Neural Net-works: Challenges and Opportunities,”
Frontiers in Neuroscience , vol. 5, 2011.[11] R. Sepulchre, G. Drion, and A. Franci, “Control Across Scales by Positive and NegativeFeedback,”
Annual Review of Control, Robotics, and Autonomous Systems , vol. 2, pp. 89–113, May 2019.[12] R. Krahe and F. Gabbiani, “Burst firing in sensory systems,”
Nature Reviews Neuroscience ,vol. 5, pp. 13–23, Jan. 2004.[13] R. M. Harris-Warrick, “Neuromodulation and flexibility in Central Pattern Generatornetworks,”
Current Opinion in Neurobiology , vol. 21, pp. 685–692, Oct. 2011.[14] E. Marder, “Neuromodulation of Neuronal Circuits: Back to the Future,”
Neuron , vol. 76,pp. 1–11, Oct. 2012.[15] R. Sepulchre, T. O’Leary, G. Drion, and A. Franci, “Control by neuromodulation: Atutorial,” in , pp. 483–497, June 2019.[16] L. Ribar,
Synthesis of Neuromorphic Circuits with Neuromodulatory Properties . Thesis,25niversity of Cambridge, 2019.[17] L. Ribar and R. Sepulchre, “Bursting through interconnection of excitable circuits,” in , pp. 1–4, Oct. 2017.[18] L. Ribar and R. Sepulchre, “Neuromodulation of Neuromorphic Circuits,”
IEEE Transac-tions on Circuits and Systems I: Regular Papers , pp. 1–13, 2019.[19] R. Sepulchre, G. Drion, and A. Franci, “Excitable Behaviors,” in
Emerging Applicationsof Control and Systems Theory , Lecture Notes in Control and Information Sciences -Proceedings, pp. 269–280, Springer, Cham, 2018.[20] G. Drion, A. Franci, V. Seutin, and R. Sepulchre, “A novel phase portrait for neuronalexcitability,”
PloS one , vol. 7, no. 8, p. e41806, 2012.[21] A. Franci, G. Drion, and R. Sepulchre, “An Organizing Center in a Planar Model ofNeuronal Excitability,”
SIAM Journal on Applied Dynamical Systems , vol. 11, pp. 1698–1722, Jan. 2012.[22] A. Franci, G. Drion, V. Seutin, and R. Sepulchre, “A Balance Equation Determines a Switchin Neuronal Excitability,”
PLoS Computational Biology , vol. 9, p. e1003040, May 2013.[23] A. Franci, G. Drion, and R. Sepulchre, “Modeling the Modulation of Neuronal Bursting:A Singularity Theory Approach,”
SIAM Journal on Applied Dynamical Systems , vol. 13,pp. 798–829, Jan. 2014.[24] A. Franci and R. Sepulchre, “Realization of nonlinear behaviors from organizing centers,”in , pp. 56–61, Dec. 2014.[25] G. Drion, A. Franci, J. Dethier, and R. Sepulchre, “Dynamic Input Conductances ShapeNeuronal Spiking,” eNeuro , vol. 2, June 2015.[26] A. Franci, G. Drion, and R. Sepulchre, “Robust and tunable bursting requires slow positivefeedback,”
Journal of Neurophysiology , vol. 119, pp. 1222–1234, Dec. 2017.[27] R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane,”
Biophysical journal , vol. 1, no. 6, p. 445, 1961.[28] J. Nagumo, S. Arimoto, and S. Yoshizawa, “An active pulse transmission line simulatingnerve axon,”
Proceedings of the IRE , vol. 50, no. 10, pp. 2061–2070, 1962.[29] C. Morris and H. Lecar, “Voltage oscillations in the barnacle giant muscle fiber.,”
Biophysical journal , vol. 35, no. 1, p. 193, 1981.[30] J. W. Moore, “Excitation of the Squid Axon Membrane in Isosmotic Potassium Chloride,”
Nature , vol. 183, pp. 265–266, Jan. 1959.[31] D. A. McCormick, “Neurotransmitter actions in the thalamus and cerebral cortex and theirrole in neuromodulation of thalamocortical activity,”
Progress in Neurobiology , vol. 39,pp. 337–388, Oct. 1992.[32] C. Beurrier, P. Congar, B. Bioulac, and C. Hammond, “Subthalamic nucleus neurons switchfrom single-spike activity to burst-firing mode,”
The Journal of neuroscience , vol. 19, no. 2,26p. 599–609, 1999.[33] S. M. Sherman, “Tonic and burst firing: Dual modes of thalamocortical relay,”
Trends inneurosciences , vol. 24, no. 2, pp. 122–126, 2001.[34] A. Kepecs, X.-J. Wang, and J. Lisman, “Bursting Neurons Signal Input Slope,”
Journal ofNeuroscience , vol. 22, pp. 9053–9062, Oct. 2002.[35] G. Marsat and G. S. Pollack, “A Behavioral Role for Feature Detection by Sensory Bursts,”
Journal of Neuroscience , vol. 26, pp. 10542–10547, Oct. 2006.[36] R. E. Plant, “Bifurcation and resonance in a model for bursting nerve cells,”
Journal ofMathematical Biology , vol. 11, pp. 15–32, Jan. 1981.[37] J. Rinzel and Y. S. Lee, “Dissection of a model for neuronal parabolic bursting,”
Journalof mathematical biology , vol. 25, no. 6, pp. 653–675, 1987.[38] E. M. Izhikevich,
Dynamical Systems in Neuroscience . MIT press, 2007.[39] E. M. Izhikevich, “Neural excitability, spiking and bursting,”
International Journal ofBifurcation and Chaos , vol. 10, no. 06, pp. 1171–1266, 2000.[40] J. L. Hindmarsh and R. M. Rose, “A model of neuronal bursting using three coupled firstorder differential equations,”
Proceedings of the Royal Society of London B: BiologicalSciences , vol. 221, no. 1222, pp. 87–102, 1984.[41] J. Dethier, G. Drion, A. Franci, and R. Sepulchre, “A positive feedback at the cellularlevel promotes robustness and modulation at the circuit level,”
Journal of Neurophysiology ,vol. 114, pp. 2472–2484, Oct. 2015.[42] G. Drion, J. Dethier, A. Franci, and R. Sepulchre, “Switchable slow cellular conductancesdetermine robustness and tunability of network states,”
PLOS Computational Biology ,vol. 14, p. e1006125, Apr. 2018.[43] G. Drion, A. Franci, and R. Sepulchre, “Cellular switches orchestrate rhythmic circuits,”
Biological Cybernetics , Sept. 2018.[44] F. Nadim and D. Bucher, “Neuromodulation of neurons and synapses,”
Current Opinion inNeurobiology , vol. 29, pp. 48–56, Dec. 2014.[45] G. B. Ermentrout and D. H. Terman,
Mathematical Foundations of Neuroscience . SpringerScience & Business Media, July 2010.[46] G. Drion, T. O’Leary, and E. Marder, “Ion channel degeneracy enables robust andtunable neuronal firing rates,”
Proceedings of the National Academy of Sciences eLS ,pp. 1–12, American Cancer Society, 2015.2750] E. Marder and D. Bucher, “Understanding Circuit Dynamics Using the StomatogastricNervous System of Lobsters and Crabs,”
Annual Review of Physiology , vol. 69, no. 1,pp. 291–316, 2007.[51] G. J. Gutierrez, T. O’Leary, and E. Marder, “Multiple Mechanisms Switch an ElectricallyCoupled, Synaptically Inhibited Neuron between Competing Rhythmic Oscillators,”
Neuron ,vol. 77, pp. 845–858, Mar. 2013.[52] G. J. Gutierrez and E. Marder, “Modulation of a Single Neuron Has State-DependentActions on Circuit Dynamics,” eNeuro , vol. 1, Nov. 2014.[53] C. Nassim,
Lessons from the Lobster: Eve Marder’s Work in Neuroscience . MIT Press,June 2018.[54] T. J. Sejnowski, “The unreasonable effectiveness of deep learning in artificial intelligence,”
Proceedings of the National Academy of Sciences , p. 201907373, Jan. 2020.[55] J. C. Maxwell, “I. On governors,”
Proceedings of the Royal Society of London , vol. 16,pp. 270–283, Jan. 1868.[56] D. Tucker, “The history of positive feedback: The oscillating audion, the regenerativereceiver, and other applications up to around 1923,”
Radio and Electronic Engineer , vol. 42,pp. 69–80, Feb. 1972.[57] A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and itsapplication to conduction and excitation in nerve,”
The Journal of physiology , vol. 117,no. 4, p. 500, 1952.[58] J. Rinzel, “Excitation dynamics: Insights from simplified membrane models,” in
Fed. Proc ,vol. 44, pp. 2944–2946, 1985.[59] T. B. Kepler, L. F. Abbott, and E. Marder, “Reduction of conductance-based neuron models,”
Biological Cybernetics , vol. 66, no. 5, pp. 381–387, 1992.[60] J. Rinzel, “A formal classification of bursting mechanisms in excitable systems,” in
Mathematical Topics in Population Biology, Morphogenesis and Neurosciences , pp. 267–281, Springer, 1987.[61] J. Rinzel and G. B. Ermentrout, “Analysis of neural excitability and oscillations,” in
Methodsin Neuronal Modeling , pp. 135–169, MIT press, 1989.[62] G. I. Cirillo and R. Sepulchre, “The geometry of rest–spike bistability,”
The Journal ofMathematical Neuroscience , vol. 10, p. 13, Dec. 2020.[63] M. Maher, S. Deweerth, M. Mahowald, and C. Mead, “Implementing neural architecturesusing analog VLSI circuits,”
IEEE Transactions on Circuits and Systems , vol. 36, pp. 643–652, May 1989.[64] M. Mahowald and R. Douglas, “A silicon neuron,”
Nature , vol. 354, no. 6354, pp. 515–518,1991.[65] R. Lyon and C. Mead, “An analog electronic cochlea,”
IEEE Transactions on Acoustics, peech, and Signal Processing , vol. 36, pp. 1119–1134, July 1988.[66] L. Watts, D. A. Kerns, R. F. Lyon, and C. A. Mead, “Improved implementation of thesilicon cochlea,” IEEE Journal of Solid-State Circuits , vol. 27, pp. 692–700, May 1992.[67] C. A. Mead and M. A. Mahowald, “A silicon model of early visual processing,”
NeuralNetworks , vol. 1, no. 1, pp. 91–97, 1988.[68] R. Douglas, M. Mahowald, and C. Mead, “Neuromorphic Analogue VLSI,”
Annual Reviewof Neuroscience , vol. 18, no. 1, pp. 255–281, 1995.[69] K. A. Boahen, “Point-to-point connectivity between neuromorphic chips using addressevents,”
IEEE Transactions on Circuits and Systems II: Analog and Digital SignalProcessing , vol. 47, no. 5, pp. 416–434, 2000.[70] N. Qiao, H. Mostafa, F. Corradi, M. Osswald, F. Stefanini, D. Sumislawska, and G. Indiveri,“A reconfigurable on-line learning spiking neuromorphic processor comprising 256 neuronsand 128K synapses,”
Frontiers in Neuroscience , vol. 9, 2015.[71] T. Yu, J. Park, S. Joshi, C. Maier, and G. Cauwenberghs, “65k-neuron integrate-and-fire array transceiver with address-event reconfigurable synaptic routing,” in , pp. 21–24, Nov. 2012.[72] P. A. Merolla, J. V. Arthur, R. Alvarez-Icaza, A. S. Cassidy, J. Sawada, F. Akopyan, B. L.Jackson, N. Imam, C. Guo, Y. Nakamura, B. Brezzo, I. Vo, S. K. Esser, R. Appuswamy,B. Taba, A. Amir, M. D. Flickner, W. P. Risk, R. Manohar, and D. S. Modha, “A millionspiking-neuron integrated circuit with a scalable communication network and interface,”
Science , vol. 345, pp. 668–673, Aug. 2014.[73] B. V. Benjamin, P. Gao, E. McQuinn, S. Choudhary, A. R. Chandrasekaran, J.-M. Bussat,R. Alvarez-Icaza, J. V. Arthur, P. A. Merolla, and K. Boahen, “Neurogrid: A mixed-analog-digital multichip system for large-scale neural simulations,”
Proceedings of the IEEE ,vol. 102, no. 5, pp. 699–716, 2014.[74] J. Schemmel, D. Briiderle, A. Griibl, M. Hock, K. Meier, and S. Millner, “A wafer-scaleneuromorphic hardware system for large-scale neural modeling,” in
Proceedings of 2010IEEE International Symposium on Circuits and Systems , (Paris, France), pp. 1947–1950,IEEE, May 2010.[75] K. Boahen, “Neuromorphic Microchips,”
Scientific American , vol. 292, no. 5, pp. 56–63,2005.[76] S.-C. Liu and T. Delbruck, “Neuromorphic sensory systems,”
Current Opinion in Neuro-biology , vol. 20, pp. 288–295, June 2010.[77] S. H. Jo, T. Chang, I. Ebong, B. B. Bhadviya, P. Mazumder, and W. Lu, “NanoscaleMemristor Device as Synapse in Neuromorphic Systems,”
Nano Letters , vol. 10, pp. 1297–1301, Apr. 2010.[78] G. Indiveri, B. Linares-Barranco, R. Legenstein, G. Deligeorgis, and T. Prodromakis,29Integration of nanoscale memristor synapses in neuromorphic computing architectures,”
Nanotechnology , vol. 24, p. 384010, Sept. 2013.[79] S. P. DeWeerth, L. Nielsen, C. A. Mead, and K. J. Astrom, “A simple neuron servo,”
IEEETransactions on Neural Networks , vol. 2, no. 2, pp. 248–251, 1991.[80] E. Marder and R. L. Calabrese, “Principles of rhythmic motor pattern generation,”
Physiological Reviews , vol. 76, pp. 687–717, July 1996.[81] T. G. Brown and C. S. Sherrington, “The intrinsic factors in the act of progression in themammal,”
Proceedings of the Royal Society of London. Series B, Containing Papers of aBiological Character , vol. 84, pp. 308–319, Dec. 1911.30
10 20 30 40 50 60 70 80 90 100050100 t [ms] V [mV]0 10 20 30 40 50 60 70 80 90 10005 t [ms] I app [ µ A / cm ] Figure 1: Input-output characteristic of excitable systems. Subthreshold inputs generate sub-threshold outputs, but suprathreshold inputs generate an all-or-none response in the form of oneor several action potentials. Response shown for the Hodgkin-Huxley neural model.31 E l g l I l Passive membrane E j g j I j Ionic currentsFigure 2: A general conductance-based circuit. The passive membrane consisting of themembrane capacitor and the leak current is interconnected with possibly many ionic currents.32 δV G L m g m L h g h L n g n Figure 3: Linearized Hodgkin-Huxley circuit. The impedance consists of the total resistivecomponent G , in parallel with the first-order branches corresponding to the gating variables.The fast positive feedback appears as the fast negative conductance branch g m − L m (red), whilethe slow negative feedback appears as the two positive conductance branches g h − L h and g n − L n (blue). All the values in the circuit are voltage-dependent, depending on the voltage point aroundwhich the linearization is considered. 33
50 100102030 GV [mV] g [ m S / c m ] Instantaneous − − g m V [mV] Fast g h g n V [mV] Slow
Figure 4: Linearized conductances of the Hodgkin-Huxley model. The conductances are groupedin three parts: instantaneous (left), fast (middle) and slow (right). The excitability stems from thecombination of the fast positive feedback (negative conductance) and the slow negative feedback(positive conductance). 34 − − − t [s] V [mV]0 5 10 15 20 25 30 35 40 45 50 − . − . t [s] I app [ µ A / cm ] Figure 5: Input-output characteristic of bursting systems. Similarly to the excitability propertyshown in Fig. 1, the subthreshold inputs generate small, subthreshold outputs. However,suprathreshold inputs generate a burst of spikes. Response shown for the Aplysia R-15 model.35 δV G L m g m L h g h L n g n L x g x L c g c Fast excitability Slow excitabilityFigure 6: Linearized bursting circuit of Aplysia R-15. The fast excitability component of thecircuit impedance consists of the same elements as the Hodgkin-Huxley linearized circuit. Theadditional slow excitability elements provide slower first-order elements that make the systemburst excitable. 36
50 0 − . − . g m g [ m S / c m ] Fast −
50 000 . . . g h g n Slow −
50 0 − − · − g x V [mV] g [ m S / c m ] Slower −
50 0 − . . · − g c V [mV] Ultra-slow
Figure 7: Linearized conductances of the Aplysia R-15 bursting model. Building from thepicture shown in Fig. 4, in addition to the fast negative and slow positive conductance necessaryfor spike generation (top), bursting requires additional slower negative and ultra-slow positiveconductances (bottom). The two slower conductances activate at lower voltages than the twofaster ones, so that the slow wave is generated in the lower voltage range. Note that at highvoltage values the ultra-slow g c becomes negative: the excursions in this range are too fast forthe ultra-slow variable to have an effect. 37 I p ( V ) I app Passive RC circuit I j Feedback currents V Figure 8: A neuromorphic neuron circuit. A capacitor and a resistive element capturing thedissipative properties of the membrane are interconnected with elementary feedback currentelements acting in different timescales. 38 I p ( v ) I app Passive membrane I − f I + s Excitability elements V Figure 9: Neuromorphic excitable circuit. The circuit consists of a parallel interconnection ofthe passive membrane components with a fast negative conductance element ( I − f ) and a slowpositive conductance element ( I + s ). 39 f I f VI Fast I-V V f V f Thresholds V f V f Range B i s t a b ili t y Amplitude
Figure 10: The N-shaped fast I-V curve. The curve represents a bistable fast system as forapplied currents in the range I app ∈ ( I f , I f ) there are two stable equilibrium voltages separatedby a middle unstable equilibrium. The range is determined by the threshold voltages V f and V f representing the points where the slope is zero. Increasing the current above I f or decreasing itbelow I f leads to a ’jump’ to the opposite branch of the curve, determined by the range voltages V f and V f respectively. The amplitude of the spikes of the full system is determined by thevoltage range ( V f , V f ) . 40 f V f Thresholds V e V e + - +Excitable Spiking VI Instantaenous I-V V f V f VI Fast I-V V e V e I app I app VI Slow I-V I app I app tI app tV Figure 11: Properties of the excitable circuit. Top: Excitability is characterized by the instanta-neous passive I-V curve, N-shaped fast I-V curve and a monotonic slow I-V curve. The slow I-Vcurve is the steady-state characteristic of the system, so that the intersection of the line I = I app and the curve determines the equilibrium of the system V e . Middle: If the equilibrium voltagelies within the negative-conductance region of the fast I-V curve, the equilibrium is unstableand the system exhibits constant spiking ( V e ). If the equilibrium voltage is outside this region,the equilibrium is stable and the system is excitable V e . Bottom: Changing the applied currentswitches the system between the two regimes. 41 f = V s I SNIC VI Slow I-V I SNIC I app Frequency V f I Hopf VI Slow I-V I Hopf I app Frequency
Figure 12: Different mechanisms of rest-spiking transition. The properties of the slow I-V curve determine the bifurcation mechanism of the transition between resting and spikingstates, giving different input-output characteristics for the spiking frequency. Top: The transitionpoint coinciding with the turning point of the slow I-V curve generates a SNIC bifurcation,characterized by a continuous frequency curve. Bottom: Monotonically increasing slow I-V curvegenerates a Hopf bifurcation at the transition point, characterized by a discontinuous frequencycurve. 42 I p ( v ) I app Passive membrane I − f I + s Fast excitability I − s I + us Slow excitability V Figure 13: Neuromorphic bursting circuit. The circuit consists of the interconnection of a passivemembrane with the feedback components generating both fast excitability ( I − f and I + s ), as wellas components generating slow excitability ( I − s and I + us ).43 f V f V s V s Thresholds+/+ +/- -/- -/+ +/+ V f V f V s V s RangeFast amplitudeSlow amplitude V f V f VI Fast I-V V s V s V s V s I s I s VI Slow I-V B i s t a b ili t y Figure 14: Slow bistability between stable resting and spiking states. Both fast and slow I-V curves of a slow bistable system are N-shaped signifying bistability. The ”up” state of theslow I-V curve corresponds to the unstable region of the fast I-V curve, so that for currents I app ∈ ( I s , I s ) the system has coexisting stable equilibrium and a stable limit cycle. The fastand slow thresholds define different regions of positive/negative conductance in the fast (givenby the first sign) and slow (given by the second sign) timescales respectively. Fast and slowbistability ranges then define the amplitudes of the fast and the slow spiking respectively of thefull bursting system. 44 f V f V s V s Thresholds+/+ +/- -/- -/+ +/+ V e V e V e Burst excitable Bursting Spiking V f V f VI Fast I-V V s V s VI Slow I-V V e V e V e I I I VI Ultra-slow I-V tVI I I tI app Figure 15: Properties of the bursting circuit. Top: Burst excitability is characterized by theinstantaneous passive I-V curve (not shown), N-shaped fast and slow I-V curves, and a monotonicultra-slow I-V curve. The ultra-slow I-V curve is the steady state characteristic of the system, sothat the intersection of the line I = I app and the curve determines the equilibrium of the system V e . Middle: If the equilibrium voltage lies outside the negative conductance regions of fast andslow I-V curves, the system is burst excitable, if it lies in the negative conductance region ofthe slow I-V curve the system is bursting, and if it lies in the negative conductance region ofthe fast I-V curve above V s the system is spiking. Bottom: The system is switched between thethree states through step changes in the input. 45 V V f V f V s V s Range tV V f V f V s V s Range
Figure 16: Controlling the bursting waveform. The bursting oscillation can be designed byconcurrently shaping the fast and slow I-V curves, thus changing the amplitudes of the fastand slow spiking. This example shows the difference between plateau (left) and non-plateau(right) oscillations by moving the negative conductance regions in the fast and slow timescaleswith respect to each other. 46 V Intraburst frequency tV tV I n cre a s i n g α + s & α − s tV Interburst frequency tV tV I n cre a s i n g α + u s Figure 17: Controlling the frequency of the oscillation. Modulating the gains of the elementsproviding positive conductance determines the frequency: increasing the gain of the slow positiveconductance element increases the intraburst frequency (left), while increasing the gain of theultra-slow positive conductance increases the interburst frequency (right). In the first case, theslow negative conductance gain is increased proportionally as well in order to keep the interburstfrequency constant. 47
V tα − s V f VI V f = V s VI Slow I-V V f VI Figure 18: Controlling the neuronal oscillation mode. The transition between bursting and slowspiking modes is determined by the magnitude of the negative conductance region of the slowI-V curve (bottom). Tracing it locally around the point V = V f = V s (middle), increasing theslow negative conductance gain (left) generates bursting, while decreasing it moves the systeminto slow spiking (right). The transition is shown at the top for a ramp input in α − s , where thedecrease in the slow bistable region gradually decreases the length of individual bursts, eventuallyleading to spiking with the disappearance of the slow bistability.48 V tV tI app I app I app Figure 19: Relay properties in the bursting and slow spiking regimes. Subjecting the neuron toa train of spikes, when the system is in the bursting mode, the fast input spikes are rejectedand the oscillating bursting behavior is uninterrupted. When the system is in the slow spikingstate, each individual spike is relayed at the output. The switch is controlled through a singleparameter corresponding to the gain of the slow negative conductance. The light blue borderaround the node represents the spiking state of the neuron.49
V tI app VI Fast I-V I app VI Slow I-V
Figure 20: Post inhibitory rebound spike in a fast bistable system. By introducing a slow negativefeedback current with a low activation to the bistable system of Fig. 10, the system exhibits arobust transient spiking response to a prolonged negative input pulse in the absence of endogenousspiking oscillation. 50
V tI app VI Fast I-V VI Slow I-V I app VI Ultra-slow I-V
Figure 21: Post inhibitory rebound burst in a slow rest-spike bistable system. Mirroring theconstruction in Fig. 20, by introducing an ultra-slow negative feedback current with a lowactivation to the bistable system of Fig. 14, the system exhibits a robust transient burst responseto a prolonged negative input pulse in the absence of endogenous bursting oscillation. This is theessential mechanism that allows networks of neurons to generate robust rhythm in the absenceof endogenous oscillatory behavior in the individual nodes.51
V tV
Figure 22: Half-center oscillator. Using the cellular mechanism described in Fig. 21, anti-phaseoscillation in a two-neuron network is achieved by interconnecting the neurons with inhibitorysynapses. When one neuron fires a burst of spikes it introduces a transient negative input currentinto the other neuron. Once this input terminates, the other neuron fires a burst of spikes inresponse, inhibiting the first neuron in alternation. If the synaptic connections are strong enough,this alternative spiking produces a robust anti-phase oscillation of the network.52 app I app tI app tV tV tI app Figure 23: Two-neuron inhibitory network switched ON. When the individual neurons are inthe bursting regime, the neurons oscillate in anti-phase defined by the inhibitory connections.The rhythm is robust to external disturbance, so that spike trains to individual neurons do notsignificantly alter the behavior. 53 app I app tI app tV tV tI app Figure 24: Two-neuron inhibitory network switched OFF. Decreasing a single parametercorresponding to the slow negative conductance gain of each neuron, neurons are switchedfrom the bursting to the spiking regime. This effectively switches the rhythm off and neuronsare able to relay individual incoming spikes, similarly to Fig. 19.54
V tV tV tV tV
Figure 25: Rhythm-generating network disconnected. The network consists of a pair of two-neuron oscillators and a middle hub neuron. The parameters are chosen so that the left oscillatoris fast and the right oscillator is slow, while the hub neuron is in the slow spiking mode.55
V tV tV tV tV
Figure 26: Rhythm-generating network connected. Hub neuron is connected to individual neuronsof the two-neuron oscillators through electrical connections. The resistive connections enable theinteraction between the two rhythms. Due to the presence of slow negative conductance in therhythm generating neurons, the bursting rhythms are robust to the input disturbances, so that thefast and slow rhythms coexist, while the hub neuron experiences a mix of the two oscillations.56
V tV tV tV tV
Figure 27: Disconnecting a rhythm from the network. By modulating the slow two-neuronoscillator through the decrease in the slow negative conductance gain, the slow cells are switchedto the slow spiking relay mode. The fast bursting rhythm effectively propagates through the hubneuron, and the network is globally oscillating with the fast rhythm.57 idebar: Multiscale in biology and engineering
Designing and controlling systems across scales is a rich and ongoing area of interestwithin the control community. Control of vehicle platoons for automated highway systems,microeletromechanical systems (MEMS), segmented large telescope mirrors, as well as bion-spired applications such as soft robotics are all examples of distributed systems where thedesired system behavior is achieved through control and sensing at the local level. Due tothe abundance of cheap actuators and sensors, there is a need for a theoretical and designmethodology for building robust complex systems with imprecise individual components. Theability of constructing robust and adaptable systems using noisy and uncertain building blocksis precisely the defining characteristic of biological systems. Understanding the organizationalprinciples that govern biological behaviors could thus inform the design of systems with suchremarkable capabilities.Control across scales is at the heart of the organization of neural networks (see Fig. S1).At the nanoscale, special proteins called ion channels open and close in a stochastic mannerand thus generate inward and outward currents that control the potential difference across acell’s membrane. At the cellular scale, the flow of ionic currents is controlled through massopening and closing of these channels, enabling the control of cell excitability. At the networkscale, the control of individual excitability properties of the cells, together with the constantmodulation of the connectivity strength between the cells, defines the pattern of the networkbehavior. These networks can function as robust and adaptable clocks that generate rhythmsfor repeatable actions such as breathing and walking, while others may filter, process and relayincoming motor and sensory signals. Finally, the interaction of many such networks enablesthe higher cognitive functions such as awareness, task selection and learning. The brain thusencompasses a vast range of spatial and temporal scales, from microseconds and nanometers ofion transport to hours and meter scales of complex brain functions.One of the main challenges associated with distributed sensing and actuating is the inherentlocal action and measurement involved: the errors at the small scales are aggregated at the largescale, making the global control through the cumulative action of individual agents a difficulttask. This does not seem to be a challenge for biological systems which retain robust globalfunction while consisting of a large interconnection of noisy, imprecise, but remarkably energyefficient components. The core of this discrepancy appears to be the difference in the feedbackstructure of the systems: engineered systems mostly involve a clear separation between positiveand negative feedback pathways, while neural systems involve interlocked mixed feedback at allscales. This enables biological networks to operate in a continuum between the purely analogand the purely digital world, a key concept behind neuromorphic design and control.58igure S1: The multiscale organization of the brain (adapted from [54]). The picture shows thehierarchy of neural organization, starting from the smallest molecular level all the way to thewhole central nervous system (CNS) that generates the behavior.59 idebar: Positive and negative feedback
Both positive and negative feedback have a long history of application in engineeringsciences, even though studying positive feedback is much less prominent nowadays in controlsystem design. From Maxwell’s pioneering work on stabilization of governors [55], negativefeedback has been recognized as the basis of robust control system design for regulation orhomeostasis. This realization came somewhat later to the world of electronics, where initially,positive feedback was utilized in order to increase the sensitivity of amplifiers, in turn bringingthe unwanted destabilizing effects [56]. Negative feedback became key to designing robust andstable amplifying circuits using unreliable components, and positive feedback was mainly utilizedas a design principle for designing memory storage units in digital electronics or autonomousoscillators. This development has lead to a clear distinction between the applications of the twofeedback loops. Positive feedback is utilized to generate bistable switches and clocks, whilenegative feedback enables robust regulation. Biological systems do not make a rigid separationbetween the negative and positive feedback pathways, but utilize both in a combined manner.Every neuronal system contains both memory and processing capabilities mixed within the samestructure, and this unique organization gives the robust, adaptable and efficient characteristicsof neural systems. Such mixed feedback principles could provide a promising new direction inengineering and control, enabling versatile control principles for multiscale designs [11].In order to illuminate the complementarity of positive and negative feedback, Fig. S2presents a simple toy example that nonetheless captures the fundamentals of the two feedbackconfigurations on an arbitrary input-output system. The simplified model of the plant is asigmoidal mapping which provides localized amplification and saturates for small and largeinputs. It models the natural saturation that occurs in any physical or biological process. Whennegative feedback is applied to the static plant, it linearizes the input-output characteristic asobserved by the extended linear region of the amplifier (left). The sensitivity of the feedbacksystem to an input change is thus decreased. On the other hand, positive feedback configurationhas the opposite effect on the input-output characteristic by increasing the sensitivity, thusreducing the linear range of the amplifier. If the positive feedback is sufficiently large, theamplifying region shrinks to zero, while the sensitivity becomes arbitrarily large around .This point of ultra-sensitivity divides the input-output behavior between two distinct regimes:a continuous amplifying regime and a bistable discrete regime. Interestingly, excitable systemsessentially operate on a continuum between these two extremes by having coexisting positiveand negative feedback loops with differing dynamical properties.In physical devices, positive and negative feedback results from suitable componentinterconnections. An elementary circuit implementation is the parallel interconnection of a60apacitor with a positive or negative resistance element (Fig. S3). Here, the effect of a singlefeedback element is considered, so that the voltage across the capacitor is given by C ˙ V = − F ( V ) , (S1)where F ( V ) is the I-V characteristics of the nonlinear resistor. The conductance of the resistoris defined as the derivative of its I-V characteristic around some voltage F (cid:48) ( V ) . The linearizedcircuit can be regarded as the negative feedback interconnection of two input-output systems:a capacitor with a transfer function /Cs and a nonlinear resistor with local gain F (cid:48) ( V ) . Thisrelationship directly relates the sign of the conductance to the sign of the feedback: positive conductance elements provide negative feedback, and negative conductance elements provide positive feedback.Whether in biophysical or engineered circuits, the static characteristic F ( V ) is onlyapproximate as it neglects the dynamics of the device. Timescale separation neverthelessallows for a quasi-static analysis of feedback loops in distinct timescales. In each temporalscale, the static characteristic then provides an estimation of the local feedback gain via thelinearized conductance F (cid:48) ( V ) . Thus, this simplified circuit interpretation is a useful aid for bothunderstanding the effect of different ionic currents in the generation of the neural behavior, aswell as for developing a synthesis methodology for the design of excitable circuits.61 ( · ) yu uy S ( · ) yu K + uyS ( · ) yu K − uy Figure S2: Feedback effect on static input-output functions. A system with a sigmoidal input-output mapping S ( u ) = tanh( u ) is connected in a negative feedback (left) and a positivefeedback (right) configuration, with the strength of the feedback connections controlled by aproportional gain K . In the negative feedback configuration, feedback linearizes the nonlinearmapping, stretching the linear region of the amplifier, while at the same time decreasing thesensitivity. On the other hand, positive feedback increases the sensitivity, and at the singularpoint, the slope becomes infinite (grey line). Increasing the magnitude of the feedback beyondthis point brings the system into the bistable regime, where the hysteretic region defines thebivalued range where the system can be either in the low or the high state.62 V VI − IV VI + IV VI IV
Figure S3: Feedback as circuit interconnection. Top represents the basic feedback configurationwhere the voltage across the capacitor is controlled by means of a single circuit element. Themiddle area represents the input-output characteristic of the initial passive element. Applyingthe input voltage V across its terminals and measuring the passing current I , the obtainedcharacteristic is a linear I-V curve for a simple resistor. Interconnecting a resistor with localizedpositive conductance (blue) and repeating the procedure produces a nonlinear, but again, purelymonotonically increasing I-V curve characterizing again a passive interconnection. On the otherhand interconnecting a resistor with localized negative conductance (red) locally decreases theslope of the measured I-V curve. At the singular point, there is a point of zero conductance (lightgray). Increasing the negative resistance of the element further generates a nonlinear N-shapedI-V curve which is a signature of bistability. 63 idebar: The Hodgkin-Huxley model The classical model of excitability comes from the pioneering experimental work ofHodgkin and Huxley (Fig. S4) of the squid giant axon in 1952 [57]. Although the analysisconcentrated on this particular neuron, the methodology was later applied to explain the behaviorof other neural cells. The starting point of the analysis is the observation that the excitablemembrane can be modeled as an electrical circuit (Fig. S5).Firstly, every cell consists of an impermeable membrane which is able to maintain anelectrical potential difference between the intercellular and extracellular environments. Thisproperty is modeled with a capacitor which stores the charge between the two media. In additionto this, the membrane is equipped with special proteins called ion channels which are selectivelypermeable to specific ions in the environment. Due to the different concentrations of ions insideand outside the cell, the cell dynamically controls its membrane voltage by opening and closingthese channels and thus controlling the flow of ionic currents through the membrane. Hodgkinand Huxley identified two key players for the generation of electrical pulses in the squid axon:potassium ( K + ) and sodium ( N a + ) ionic currents. Remaining currents were lumped into a third,leak component. The selective permeability to each ion is captured by an individual conductanceelement in the circuit, while the equilibrium (Nernst) potential where diffusion exactly balancesthe electrical force is modeled by a battery. There is a higher concentration of sodium outsidethe cell than inside, and vice-versa for potassium. Within the circuit modeling framework, thistranslates to a high sodium Nernst potential (
40 mV in the Hodgkin-Huxley model) and a lowpotassium Nernst potential ( −
70 mV in the Hodgkin-Huxley model). Thus, sodium current isalways inward (negative by convention) and acts to increase the voltage, while potassium currentis outward (positive by convention) and acts to decrease the voltage.The dynamics of the circuit are governed by the membrane equation
C dVdt = − g l ( V − E l ) − g Na ( V − E Na ) − g K ( V − E K ) + I app , (S2)where C is the membrane capacitance, V is the membrane voltage, g l , g Na , and g K are theconductances corresponding to leak, sodium, and potassium respectively, with E l , E Na and E K being their corresponding equilibrium potentials, and I app is the externally injected current intothe cell.Sodium and potassium conductances are active i.e. they are voltage and time-dependent.This reflects the continuous opening and closing of the their ion channels, in contrast to theleak conductance which is constant, and accounts for the passive properties of the membrane.By using the technique known as voltage clamping , Hodgkin and Huxley were able to keep themembrane voltage fixed at different levels and measure different step responses from the resting64tate of the system. Fitting the data to the simplest form, they obtained the equations for thetwo conductances: g Na = g Na m h, (S3) g K = g K n , (S4)where g Na and g K are the maximal conductances of sodium and potassium respectively, and m , h , and n are the gating variables that follow first-order dynamics τ m ( V ) ˙ m = m ∞ ( V ) − m,τ h ( V ) ˙ h = h ∞ ( V ) − h,τ n ( V ) ˙ n = n ∞ ( V ) − n. (S5)Each gating variable has a value between 0 and 1 and thus represents the continuous tuning ofthe ion channels between being fully closed and fully open. The steady-state functions m ∞ ( V ) , h ∞ ( V ) , and n ∞ ( V ) have a sigmoidal shape (Fig. S6, left), while the voltage-dependent timeconstants τ m , τ h , and τ n have a Gaussian shape (Fig. S6, right).The dynamics of each gating variable is fully characterized by these two voltage-dependentfunctions. The steady-state functions are monotonic and the voltage range in which the slopeof these functions is non-zero defines the window in which the currents are active, while theslope defines the sign of the feedback. The voltage-dependent time constants define the timescalewindow in which the currents operate. 65igure S4: Andrew Huxley (left) and Alan Hodgkin (right).66 E l g l I l E Na g Na I Na E K g K I K I app Intracellular mediumExtracellular medium V Figure S5: The Hodgkin-Huxley circuit. The neural membrane, separating the intracellular andthe extracellular media, is modeled as a parallel interconnection of the passive capacitor andleak current I l , together with the active sodium I Na and potassium I K currents. External currentapplied to the cell is represented with the current source I app .67
50 0 50 10000 . . . . V [mV] m ∞ ( V ) h ∞ ( V ) n ∞ ( V ) −
50 0 50 10002468 V [mV] τ [ms] τ m ( V ) τ h ( V ) τ n ( V ) Figure S6: Steady-state and time constant functions of the gating variables. The steady-statefunctions (left) have a sigmoidal shape and are monotonically increasing for activation variables,and monotonically decreasing for inactivation variables. The time constants (right) have aGaussian shape. 68 idebar: Linearizing conductance-based models
Conductance-based models can be analyzed by considering small perturbations aroundevery equilibrium voltage [25]. This technique characterizes the circuit based on how itsimpedance function changes with the equilibrium voltage and frequency, illuminating importantproperties of possibly highly nonlinear and complex systems that are generally hard to analyze.When a small change in the membrane voltage is applied, it induces a current change thathas two components: an instant change in the current, due to the resistive properties of each ioniccurrent, and a slower component, due to the active properties of the gating variables. In circuitterms, the former leads to a simple resistor that gives an instantaneous change in current fora change in voltage, and each gating variable leads to a resistor-inductor branch, capturing theslower, first-order dynamics that arise. The value of each component will depend on the voltagearound which the linearization is applied. Applying this to an ionic current with conductance asin (2) gives the change in current δI j = ∂I j ∂V δV + ∂I j ∂m j δm j + ∂I j ∂h j δh j , (S6)where the first term represents the resistive component of the current, while the second and thethird term are due to the activation and inactivation function. Taking the Laplace transform, thefull linearized membrane model thus becomes: sCδV = − (cid:20) G ( V ) + (cid:88) j (cid:16) g m j ( V ) τ m j ( V ) s + 1 + g h j ( V ) τ h j ( V ) s + 1 (cid:17)(cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Z ( s, V ) δV. (S7)The behavior of the linearized membrane model is fully characterized by the capacitance C and the total impedance Z ( s, V ) which consists of the total resistive part G ( V ) accumulatingall resistive components of the ionic currents G ( V ) = g l + (cid:88) j ∂I j ∂V , (S8)and the first-order terms arising from the dynamics of the gating variables g m j ( V ) = ∂I j ∂m j dm j, ∞ ( V ) dV , (S9) g h j ( V ) = ∂I j ∂h j dh j, ∞ ( V ) dV . (S10)Linearization of ionic currents has a simple circuit implementation shown in Fig. S7, withthe single resistor capturing the resistive term G j ( V ) , and a series connection of a resistor and69n inductor capturing the first order terms due to the activation and inactivation functions. Thisinterpretation illuminates the synthesis problem, as it decouples the input-output properties ofthe circuit elements from their internal physical realization. Linearization is a indispensable toolfor tractable analysis of excitable behaviors as it allows the effects of the dynamic variablesto be separated, and contributions from many different ionic currents to be grouped togetheraccording to their temporal range of activation.70 j g j I j G j L m j g m j L h j g h j δI j Figure S7: Linearization of an arbitrary ionic current with an activation and an inactivationvariable. The resistive term is represented by a single resistor G j , while each of the serialconnections of a resistor and an inductor represents the terms stemming from the activationfunction m j and the inactivation function h j . 71 idebar: I-V curves and phase portraits Classically, the mathematical analysis of excitability has centered around the phase portraitanalysis of dynamical models. Phase portraits provide a geometrical understanding of excitabilitythanks to its fundamentally two-timescale nature. The famous FitzHugh-Nagumo model [27]provides the core illustration of this property through its two state variables: a voltage variablecapturing the fast positive feedback process, and a slow refractory variable capturing the slowrecovery. The model provides key insight into the generation of an action potential through theanalysis of the nullclines and the equilibrium analysis of the system. The inverted N-shape ofthe fast nullcline reflects the bistable nature of the fast subsystem, while the monotone slownullcline ensures the monostability of the system in the slow timescale. Remarkably, reductionof more complex spiking models reveals the same excitability picture [58], [59], showing thegenerality of the mechanisms discussed by FitzHugh in his seminal paper.Richer multi-timescale phenomena such as bursting are more difficult to capture with atwo-dimensional phase plane [38]. Still, because of their ease of interpretability, two-dimensionalphase portrait techniques were also successfully utilized to understand more complex neuronalbehavior [23], [39], [40], [60]. Those models augment the phase plane analysis of spiking witha parameter that deforms the phase portrait in a slower timescale.Traditionally, phase plane models have been derived from empirical reductions of higher-dimensional conductance-based models. This is not necessarily an easy task, as many distinctcurrents and parameters control neuronal excitability. Input-output techniques offer an attractivealternative to state-space methods, closer in spirit to the experimental techniques of electro-physiology. Input-output measurements are directly generated through experiments using theclassical voltage-clamp technique [57]. The technique utilizes an experimental setup using ahigh gain negative feedback amplifier that allows fixing the membrane voltage at differentlevels and measuring the total internal current, thus capturing the step response of the system.Secondly, input-output characterization naturally leads to a multiscale modeling approach as thecharacterization can be captured at different timescales, corresponding to the timescales of thespiking behavior [25]. This is the idea behind the I-V curve system analysis.The classical phase portrait pictures can be linked to the input-output characteristic of thesystem in a simple way. This is shown in Fig. S8. The figure links the standard phase portraitsof excitability with their corresponding characteristic fast and slow I-V curves. The two spikingmodels presented are FitzHugh-Nagumo and Morris-Lecar [29], representing two different typesof common spiking behavior. In the case of FitzHugh-Nagumo, the model undergoes a Hopfbifurcation at the onset of spiking, leading to the oscillations appearing with a non-zero frequency.72n the other hand, the parameters of the Morris-Lecar can be set up so that it represents a minimalrealization of a spiking neuron with continuous frequency curve [61], characterized by a saddle-node on invariant circle (SNIC) bifurcation at the onset of spiking. Both models share the sameN-shaped fast I-V curve, stemming from the existence of bistability in the fast timescale. Theydiffer in the monotonicity of the slow I-V curve: a monotonically increasing slow I-V curve isa signature of the Hopf bifurcation, while the non-monotonic slow I-V curve gives rise to theSNIC bifurcation.Two representative phase portraits of burst excitability are shown in Fig. S9. The keycharacteristic of bursting systems is the appearance of bistability between stable rest and spikingstates, which allows slower processes to periodically switch the system between the two. The twobursting phase portraits display the different mechanisms of achieving the bistability betweenthese two states. The left phase portrait is the well-known Hindmarsh and Rose model [40],where the slow nullcline is non-monotonic. The non-monotonicity of the slow variable reflectsthe appearance of both positive and negative feedback processes in the slow timescale. At theright-hand side is the phase portrait of the more recent bursting model organized by a transcriticalbifurcation [23], where bistability is instead achieved by mirroring the fast nullcline, again theconsequence of the appearance of both positive and negative slow feedback. Both mechanismslead to a robust generation of bistability between resting and spiking states of the model.Interestingly, both of these models share the same qualitative picture in their input-output I-V curve characterization: the N-shaped fast I-V curved is accompanied by the N-shaped slowI-V curve, a signature of the appearance of slow bistability between the two states. A moreextensive analysis of the geometry of rest-spike bistability is provided in [62].We note that the phase portraits illustrated here do not include two classical bistable phaseportraits that lack a source of slow positive feedback. Both phase portraits of Fig. S8 can exhibitthe coexistence of a stable fixed point and a stable limit cycle through the appropriate tuningof the timescales in the models and have often been associated to bursting in neurodynamicalstudies [61]. The models are however bistable only for a precise ratio of timescales and do notcapture the robust transitions between slow spiking and bursting. These limitations are furtherdiscussed in [22], [26], as well as in terms of the neuromorphic implementation in [18].73 n FitzHugh-Nagumo VI Fast I-V I app VI Slow I-V Vn Morris-Lecar VI Fast I-V I app VI Slow I-V
Figure S8: Phase portraits of excitability and their corresponding I-V curves. The FitzHugh-Nagumo phase portrait consists of an inverted N-shaped fast V nullcline and a linear slow n nullcline. The single equilibrium of the system loses stability through a Hopf bifurcation nearthe local maxima, leading to finite frequency oscillations at the onset. This is reflected in theI-V curves as an N-shaped fast I-V curve, and a monotonic slow I-V curve. On the otherhand, Morris-Lecar model can exhibit the loss of stability through a saddle-node on invariantcircle (SNIC) bifurcation, observed through the appearance of three nullcline intersections: astable equilibrium on the left, a saddle point in the middle, and an unstable equilibrium on theright. While the fast I-V curve assumes a similar N-shape, the SNIC bifurcation mechanism ismanifested through the non-monotonicity of the slow I-V curve.74 n Hindmarsh-Rose VI Fast I-V I app VI Slow I-V Vn Transcritical VI Fast I-V I app VI Slow I-V
Figure S9: Phase portraits of rest-spike bistability and their corresponding I-V curves. Theessential mechanism behind burst excitability is the bistability between a stable rest state and astable spiking state. In Hindmarsh-Rose model this is achieved through the quadratic form ofthe slow nullcline, so that three intersection points give rise to a stable equilibrium on the left, asaddle point in the middle, and the unstable equilibrium on the right, surrounded by a stable limitcycle. In the more recent transcritical model, this bistability is achieved instead by mirroringthe fast cubic nullcline, giving rise to the same three equilibrium structure. Both mechanismsshare qualitatively the same I-V curves, the fast N-shaped I-V curve due to the presence of fastbistability, and the N-shaped slow I-V curve due to the presence of slow bistability. The importantcharacteristic of both models is the presence of slow positive feedback. This is showcased bythe appearance of the negative slope region of the slow I-V curves that does not correspond tothe negative slope region in the fast timescale, which necessarily comes from a source of slownegative conductance. 75 idebar: Three decades of neuromorphic engineering
The foundations for the neuromorphic approach to designing electrical circuits were laid outin the pioneering work of Carver Mead (Fig. S10) and colleagues in the late 1980s [1], [8], [63].By making an analogy between the behavior of MOS transistors at low operating voltages and thechannel dynamics of neurons, they provided a novel methodology for synthesizing bioinspiredsystems by using the physics of the devices as a computational resource. This analog wayof computing is in contrast with the established digital technology where transistors are purelyviewed as on/off switches, and computation is achieved by abstracting their behavior and utilizingthe principles of Boolean algebra.One of the main advantages of computing in this low-voltage analog regime is the energyefficiency. By making the currents orders of magnitude smaller than in conventional digitalelectronics, neuromorphic circuits are able to naturally compute continuous operations such asexponentiation, multiplications and summations using little power. This discovery has highlightedthe potential of low-power analog circuits in emulating the efficiency and efficacy of biologicalsystems in silicon chips.Initial studies into neuromorphic architectures led to the first developments of electricalcircuits emulating the structure and operation of neurons, sensory organs and the fundamentalorganizational principles of neural networks. These included the first developments on imple-menting the conductance-based structure of neurons, which led to the first silicon Hodgkin-Huxley based neuron [64], as well as replicating the auditory and vision sensory systemsthrough the silicon cochlea [65], [66] and the silicon retina [67]. All of these devices aimedto mimic the analog computation that is achieved by biological sensors, thus drasticallyreducing the redundancy in the sensory information collected. Apart from computational,novel communication methods were developed to circumvent the inability of implementingthe massively interconnected neural structures found in biology. This was achieved by cleverlycombining the analog processing of individual neurons with digital communication through acentral communication hub in the address-event representation [68], [69].The initial research into neuromorphic computing has since influenced many otherapproaches for developing neuroinspired hardware. Several neuromorphic chips have beendeveloped [6], such as [70], [71], as well as the larger-scale IBM True North [72], Neurogrid[73] and BrainScaleS [74]. All of these use different levels of abstraction, as well as a differentdigital/analog mix for implementing the neural computations. A significant advance has beenmade in the area of neuromorphic vision, allowing for commercial products that use the retinalprinciples [75], [76]. Recent developments into novel devices such as memristors, together with76he rapid development of artificial neural networks, has directed research into finding ways ofusing these devices as natural implementations of synaptic neural interconnections that wouldself-adapt and learn [77], [78]. Future advancements in this direction may lead to intelligentsystems that would have learning and adaptation capabilities present on the physical level,providing engineering solutions with possibly unparalleled efficiency.Neuromorphic engineering provides a promising approach to utilize biological principlesin designing electrical hardware. The techniques used do not necessarily strive for accuratebiological inspiration, but use bioinspiration in designing novel circuit architectures [4].A major bottleneck of neuromorphic engineering comes from the transistor mismatch [8],[9], which results in large uncertainty in the activation range of the small currents. The presentarticle aims at highlighting the mixed feedback structure at the core of biological neurons andthe potential of exploiting this feedback architecture to compensate for the uncertainty inherentto multiscale systems. 77igure S10: Carver Mead.78 idebar: The first neuromorphic controller
An early example illustrating the potential of neuromorphic architectures in tacklingmultiscale control can be traced to [79], where the authors presented the first neuromorphicdesign of a servo controller.The structure of the controller is shown in Fig. S11. The output of a conventionalproportional-derivative (PD) action controller is put through a neuronal circuit which convertsthe output of the controller into a pulse train. The output is differential so that the positive andnegative values of the output are sent through two independent pulse channels. The controlleroutput is then fed into a conventional DC motor.The speed regulation of the neuromorphic controller can be seen in Fig. S12. Theremarkable aspect of the controller can be observed by comparing its operation at a high speedof rotation and at a low speed. At medium and high speeds both the conventional analog and theneuromorphic controller behave reliably: the neuromorphic controller effectively operates witha pulse-width modulation scheme, as the average number of pulses per second determines theDC value supplied to the motor. However, lowering the speed set point to a significantly smallervalue, the conventional analog controller fails to rotate the motor due to its inability to overcomestatic friction. In sharp contrast, the neuromorphic controller naturally adapts its operation to ascheme resembling the operation of a stepper motor: each individual pulse has enough energyto overcome the friction, and therefore the controller provides repeating kicks to the motor thatallow it to rotate at a small average frequency (Fig. S12, right).This work came about through the collaboration of, among others, Karl Astrom andCarver Mead, pioneers in control and neuromorphic engineering respectively. It highlights therevolutionary potential of neuromorphic control. The mixed analog-digital operation of neuralsystems allows for novel adaptation schemes not possible in the purely analog or the purelydigital world. 79igure S11: Neuromorphic proportional-derivative (PD) motor controller. Differential output of aconventional PD circuit is passed through two neuron-like pulse generators. Adapted from [79].80igure S12: Speed control of the neuron servo. The neural controller provides a pulse-widthmodulation control scheme at high speeds, and a stepper control scheme at lower speeds. Theright figure provides the zoomed in time plot of the low speed regulation. Adapted from [79].81 idebar: Using physics to compute: subtreshold electronics
The fundamental idea behind the neuromorphic approach pioneered by Carver Mead andcolleagues was using physics to compute, i.e. using the native relationships between currentsand voltages in electronic devices in order to implement computational primitives. At the coreof this approach is the use of standard metal-oxide-semiconductor (MOS) technology operatingin the low-voltage subthreshold regime.A transistor is a three-terminal current source element (Fig. S13): the current going betweenits drain and its source (output current) is controlled by the voltage applied between its gate andsource (input voltage). In traditional digital electronics, MOSFET is only considered on if theinput voltage is sufficiently high, disregarding the small output current when the input is belowthe threshold voltage. In this regime the output current is a quadratic function of the input voltageand the charge is mainly carried by drift. In turn, in the subthreshold regime the output current isorders of magnitude smaller and there is an exponential input-output relationship. In this region,the charge is mainly carried by diffusion and the transistor’s input-output characteristic can bereduced to the form i out = i e vinv , (S11)where i is the zero-bias current, an inherent property of the transistor, and v is the voltageconstant depending on the properties of the integrated design process and the operatingtemperature.There are two main advantages of operating in this regime: • The voltages and the currents are low. This leads to circuits operating with very low powerrequirements in the order of µ W s. • The exponential input-output relationship is a powerful analog primitive for synthesizingsigmoidal activation functions.Analog subthreshold circuits compute fundamental functions such as the exponential andlogarithmic function, as well as standard multiplication, division, addition and subtraction in anexceptionally energy efficient way, and thus come close to emulating the way biological systemsachieve analog computation. They offer an exciting avenue for designing novel neuromorphiccontrol schemes for mixed analog-digital excitable circuits.82 DS v in i out v in i out Figure S13: An n-channel MOSFET. The output current i out between its drain (D) and source (S)is controlled by the input voltage v in between its gate (G) and source, and is largely independentof the voltage between its output nodes. In the subthreshold regime, the input-output relationshiptakes an exponential form. 83 idebar: Neuromorphic building blocks Two basic subthreshold circuit building blocks are sufficient for the design of neuromorphiccircuits using the I-V curve synthesis methodology. These are the transconductance amplifier,a device that generates a sigmoidal input-output relationship, and the integrator-follower circuitthat provides temporal integration.The transconductance amplifier generates an output current that depends on the differencebetween the two input voltages. The schematic of the differential amplifier is shown in Fig. S14.In the subthreshold regime, the input transistors generate currents i and i that are areexponentially related to the two input voltages. The total sum of the two currents is controlledthrough the bottom bias transistor that provides the control current. These two currents aresubtracted to form the output current through the current mirror formed by the two uppertransistors which copies the current i into the output node. This simple circuit architectureconveniently generates a hyperbolic tangent mapping i out = i b tanh (cid:18) v − v v (cid:19) , (S12)where v is the voltage constant depending on the integrated process parameters and the circuittemperature.The circuit provides a versatile building block for the synthesis of I-V curves, as it provideslocalized conductance characteristic whose gain can be controlled externally through the input v b over several orders of magnitude. These two characteristics make it a sufficient building blockfor defining the local action of the positive and negative feedback elements of the neuromorphicarchitecture.The same element can be used to generate temporal filtering. Since the time constants ofneuromorphic systems generally require high resistor values in order to align with the frequenciesat which biological systems operate, the circuit can be utilized as a resistive element with variableresistance. This is shown in Fig. S15.The output voltage of the circuit is determined by C dv out dt = i b tanh (cid:18) v in − v out v (cid:19) . (S13)For small changes in the voltage the equation takes the form of a linear first-order filter, and thetime constant of the circuit can be appropriately defined as T = 2 v i b C. (S14)84he circuit therefore provides an effective dynamical primitive for defining the time constantsof the neuromorphic feedback elements. 85 b v v v b i out i i i V DD = − + v b v v i out Figure S14: Transconductance amplifier. The circuit realizes a hyperbolic tangent mapping fromthe differential voltage input to the current output. The gain of the function is determined bythe current flowing through the base transistor ( i b ), controlled by its base voltage ( v b ) that actsas the control input to the amplifier. 86 + v in v out v b C Figure S15: Follower-integrator circuit. The circuit implements nonlinear first-order filtering ofthe input voltage v in . The time constant of the filter can be controlled by changing the voltage v b , which effectively controls the output resistance of the amplifier.87 idebar: A neuromorphic neuron in hardware A neuromorphic neuron obeying the control principles of this article was realized inhardware as part of the PhD dissertation [16]. The circuit possess the interconnection architectureintroduced in ”Synthesis and control of neuromorphic circuits” and has a simple implementationusing the standard neuromorphic building blocks. An individual feedback element is implementedas a series interconnection of a follower-integrator circuit providing the first-order filtering, anda transconductance amplifier, providing the hyperbolic tangent input-output relationship. Thisinterconnection is shown in Fig. S16, and the details of the individual circuits are discussed in”Neuromorphic building blocks”.The main control parameters of the element then correspond to the voltage v τ that sets thetimescale of the element, the offset voltage variable v δ , and the gain of the feedback element i b set through v b . These parameters correspond to the dimensionless parameters discussed in”Synthesis and control of neuromorphic circuits” through simple transformations i b = α (cid:0) Gv (cid:1) , (S15a) v δ = δ (cid:0) v (cid:1) , (S15b)where v is the voltage constant depending on the integrated process and the circuit temperature,and G defined as the conductance of the passive membrane element around equilibrium G = di p ( v ) dv (cid:12)(cid:12)(cid:12)(cid:12) v = v e . (S16)The conductance of the passive element and the membrane capacitor define the timeconstant of the membrane equation T v = CG , (S17)and the applied current is simply derived from the dimensionless applied current as i app = I app (cid:0) Gv (cid:1) . (S18)A measurement from the hardware realization is shown in Fig. S17 capturing the continuousmodulation of the circuit between spiking and bursting regimes through the sole control of thevoltage that defines the gain of the slow negative conductance. This switch drastically changesthe input-output behavior of the circuit, thus enabling the localized control of the network outputwhen the neuron is part of a larger interconnected structure.88 + x v = − + v τ C − + v δ v x i + x v b i = 0 v Figure S16: Implementation of a single circuit feedback element. The first transconductanceamplifier and a capacitor form a nonlinear first-order filter. The output of the filter is then fedinto the second transconductance amplifier which forms the output current i + x . The figure showsa realization of a positive conductance element, while for a negative conductance element theinputs to the second amplifier are interchanged.89igure S17: Transition between slow spiking and bursting in the neuromorphic circuit implemen-tation. A periodic triangle wave is applied to the base voltage of the slow negative conductance(blue trace) so that the circuit is periodically moving between the bursting and slow spikingregimes. 90 idebar: Central pattern generators Generating robust and flexible rhythmic activity is necessary for a variety of essentialbiological behaviors such as locomotion and breathing. Special neural networks called centralpattern generators (CPGs) are responsible for generating autonomous rhythmic patterns. Thesepatterns then define spatio-temporal sequences of activation that generate synchronized motorbehavior. CPG networks vary greatly in structure and complexity [80], but share a commonmodular organization. The fundamental module in this organization is a simple two-neuronnetwork commonly known as the half-center oscillator .The half-center oscillator finds its origins in the seminal work of Brown [81], whoinvestigated the autonomous electrical activity of electro-stimulated muscles of dead animals.The simplest model of a half-center oscillator consists of two neurons mutually interconnectedwith inhibitory synaptic connections. The individual neurons do not oscillate in isolation butthe network rhythm emerges from the interaction. The self-regenerative feedback necessaryfor sustained oscillatory behavior is provided through the post-inhibitory rebound mechanismdiscussed in the main article.The mutual inhibition motif appears to be a fundamental building block shared by allrhythmic neural network. 91 uthor Biographyuthor Biography