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...er representation of fractional numbers using a limited number of bits. Fixed point math is a requirement where processors do not have floating point units, such as cell phones and embedded platforms =-=[55]-=-. Just as the name implies, the decimal point is always fixed and cannot change during runtime in comparison to floating point numbers. Fixed point representations can only store a limited range of va...

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...r the many ionic currents across the CHAPTER 1. INTRODUCTION 3 cellular membrane. Consists of 4 differential equations (expanded to over 7 by later researchers [12]) and has many tunable coefficients =-=[13]-=-. 1.2 Synapses Communication and coordination between neurons is made possible by synapses. When a synapse connects two neurons together the postsynaptic neuron receives the ionic curent while the pre...

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... the gate [41]. These resistors must be large to limit current consumption, their values are not critical but should closely match. Pinch resistors would be a good candidate for a CMOS implementation =-=[60]-=-. Figure 5.6: Circuit diagram of the nonlinear load. 5.4.3 Connection of the first two blocks It is the shape of this nonlinear load that gives the overall shape to each spike. As the voltage across t...

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... Van der Pol’s equation of a relaxation oscillator. Cannot output tonic bursting. Created as a simpler representation of excitable-oscillatory outputs compared to the more complex HodgkinHuxley model =-=[10]-=-. Morris-Lecar Based on voltage oscillations observed in barnacle muscle fibers. An accurate bio-physical model, meaning each variable has a biological connection. Output is capable of spiking but not...

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...uation is (3.13) where fclk is the switching frequency and Ca is the switched capacitor value in Farads. Generally, the switching frequency must be 100 times greater than the highest signal frequency =-=[41]-=-. This is because the equivalent resistance is based on the assumption that the switching time is considered negligible. Another advantage of the switched capacitor integrator is that the coefficient ...

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... An accurate bio-physical model, meaning each variable has a biological connection. Output is capable of spiking but not bursting. Consists of two differential equations, based on a conductance model =-=[11]-=-. Hindmarsh-Rose The selected neural model for implementation, description and analysis of this model is given in chapter 2. Hodgkin-Huxley One of the most complex conductance models and considered th...

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...xx3= Ts c·x2ms·Cx CHAPTER 3. HINDMARSH-ROSE ANALOG COMPUTERS 19 3.4.1 Switched Capacitor Integrators Switched capacitor integrators can overcome the intrinsic limit on the sizing of resistors in CMOS =-=[40]-=-. A switched capacitor integrator is shown in figure 3.3. The overall function of the switched capacitor integrator is similar to equation (3.2). The only difference is that resistor R is replaced by ...

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...ss. Despite being simpler with less governing equations and coefficients, the HR model is accurate to neurons seen in biology and was created to accurately follow the bursting output seen in mollusks =-=[28]-=-. Discrete electronic versions have also been compared to those seen in the crustacean stomatogastric system where a high degree of similarity was observed [29]. Electronic versions have even replaced...

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...isting of only a single differential equation. A summation of currents charges (or discharges) a capacitor. When a threshold voltage is reached the capacitors voltage is reset and the process repeats =-=[6]-=-. It is limited in the variety of outputs it can produce, usually only to tonic spiking. Another variation of this model type is the self-resetting neuron described by Carver Mead [7] and a similar mo...

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... change during runtime in comparison to floating point numbers. Fixed point representations can only store a limited range of values but the arithmetic is extremely quick compared to utilizing floats =-=[56]-=-. A Q4.12 representation was chosen: 4-bits for whole numbers and 12-bits for the fractional part. Table 4.2 show each bits deciamal representation for this format. This is because the DTM model’s max...

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... ideal current source was used. 5.4.4 Adaption Integrator The adaption integrator is based on a transconductance amplifier. The current output is proportional to the difference between the two inputs =-=[61]-=-. The capacitor, Cz, is then charged or CHAPTER 5. REDUCED COMPLEXITY HINDMARSH-ROSE CIRCUIT 48 discharged by the current from the amplifier creating the output voltage Vz. A capacitors current is pre...

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...et. Resonate and Fire Exhibits more dynamical output compared to the previous model but cannot produce tonic bursting. Consists of a single equation (using complex variables) or two simpler equations =-=[9]-=-. FitzHugh-Nagumo Consists of two coupled differential equations and based on Van der Pol’s equation of a relaxation oscillator. Cannot output tonic bursting. Created as a simpler representation of ex...

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.... . . 35 4.4 Example of multiplying fixed point numbers . . . . . . . . . . . . . . . . . . . . . 36 v Chapter 1 Introduction Research in the biomimetic robot field has increased over the past decade =-=[1]-=-. State machine control of robots may perform poorly in adverse conditions due to a large number of edge cases, because only a limited number of sequences can be programmed or anticipated for. Mimicki...

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...neuron operation. For the Hindmarsh-Rose, the author states a low end of 120 operations per millisecond of operation. An alternative to numerical integration is to use discrete time–map based neurons =-=[49]-=-. In depth analysis and application of this digital neuron is the subject of this chapter. Another advantage to using digital neurons is the ease of analysis of large neural maps [49] and the ease of ...

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...Its construction and analysis is similar to that of implementations of the HodgkinHuxley neural model [18] [13]. The original two dimensional HR equations were modeled after the Fitzhugh neural model =-=[34]-=-. The third dimention was added later to include triggered bursts in order to more accurately model neurons seen in nature. 2.3.1 Equilibrium Analysis & Spiking Mechanism Equations (2.1) and (2.2) are...

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...the Hindmarsh-Rose Neural Model 2.1 Introduction and Biological Background The Morris-Lecar [11] and Hodgkin-Huxley [13] models try to replicate the electrophysiological process of biological neurons =-=[27]-=- based on conductance models with many governing equations and coefficients. The Hindmarsh-Rose(HR) model is based on the global behavior of the neuron and its underlying operation is removed from the...

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...n the variety of outputs it can produce, usually only to tonic spiking. Another variation of this model type is the self-resetting neuron described by Carver Mead [7] and a similar model in reference =-=[8]-=-. In both, a capacitor is charged until a threshold voltage is reached at which point an output spike is produced and the capacitor voltage is reset. Resonate and Fire Exhibits more dynamical output c...

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...shown to be 0.2 µS, which is orders of magnitude faster than its biological equivalent due in part to the sizing of the 60 pF capacitor. Reference [20] proposes a neuron based on the Izhikevich model =-=[21]-=-. The burst length reported is on the order of a few microseconds, which is much faster than the hundreds of milliseconds of its biological counterpart. This increase in speed is suitable for computat...

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... that can operate wireless in three dimensions and have life-like movement [22]. Another is a robot snake that can be used to transverse harsh obstacles such as a debris field to search for survivors =-=[23]-=-. Researchers at the Northeastern University Marine Science Center have created several iterations of CPG based Lobster and Lamprey robots (shown in figure 1.1) for aquatic mine and object detection [...

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...]. Researchers at the Northeastern University Marine Science Center have created several iterations of CPG based Lobster and Lamprey robots (shown in figure 1.1) for aquatic mine and object detection =-=[24]-=- [2]. Neuromorphic sensors were used as environmental sensors. The bodies of the robots themselves were constructed to be as close as to nature as possible. Artificial muscles were created with nitino...

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...identifiable and accessible using in vitro analysis. This has allowed researchers to make a one-for-one correspondence between an identified neuron or central pattern generator to a behavioral output =-=[3]-=-. Of the many neural models created to replicate neural membrane potential, the Hindmarsh-Rose model was chosen for application in the novel micro-robot, the Cyberplasm. This model was analyzed and im...

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...c system where a high degree of similarity was observed [29]. Electronic versions have even replaced neurons removed from a central pattern generator of a live lobster and have restored functionality =-=[30]-=-. 2.2 Three Coupled Differential Equations The three coupled equations of the Hindmarsh Rose neural model are shown in (2.1), (2.2) and (2.3). Reference [31] has added a fourth equation, w, to the HR ...

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...ial Despite consisting of a simple update map, the DTM neural model can exhibit a wide variety of neural outputs. An expanded DTM neuron that includes sub-threshold oscillations is shown in reference =-=[52]-=-. The Discrete time–map based neuron consists of two variables: a fast (x) variable and a slow (y) variable. (4.1)-(4.3) are the governing equations of the DTM model. Coefficient µ is usually set to 0...

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...ve lobster and have restored functionality [30]. 2.2 Three Coupled Differential Equations The three coupled equations of the Hindmarsh Rose neural model are shown in (2.1), (2.2) and (2.3). Reference =-=[31]-=- has added a fourth equation, w, to the HR equation set that models 8 CHAPTER 2. HINDMARSH-ROSE NERUON 9 extremely slow dynamics (such as calcium exchanges). The fourth equation expands regions where ...

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...ing is proposed in [19]. The time between chewing is shown to be 0.2 µS, which is orders of magnitude faster than its biological equivalent due in part to the sizing of the 60 pF capacitor. Reference =-=[20]-=- proposes a neuron based on the Izhikevich model [21]. The burst length reported is on the order of a few microseconds, which is much faster than the hundreds of milliseconds of its biological counter...

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...re found by mechanical look-up tables. The most well known application was the firing computer used on naval ships which was capable of pivoting a large cannon, while moving, to track a moving target =-=[36]-=-. Electronic analog computers operated in a similar manor, did not suffer from mechanical fatigue, and gave more precise answers. Created from networks of summing circuits, integrators and trim pots f...

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...apton substrate. The muscles will be coordinated to underlie forward movement by articulating an anatomical tail. The overall micro-robot is to be powered by a chemical battery created using bacteria =-=[26]-=-. The entire system will be tied together with an analog implementation of a biologically realistic nervous system. Multiple electronic CPGs will coordinate the output muscles to create a rhythmic swi...

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... relationship in an analog computer. Reference [31] gives an overview of the implmentation in a discrete case but does not expand on scaling factors involved or provide a schematic diagram. Reference =-=[39]-=- covers an implemen14 CHAPTER 3. HINDMARSH-ROSE ANALOG COMPUTERS 15 tation into 0.13 µm CMOS but ignores the biological time scales relevant for robotic operation. Section 1.4 covers the time scales s...

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...controller would then pass timing control parameters to each leg CPG to coordinate walking or running [16]. Insects contain the simplest networks and can be easily studied and adapted for robotic use =-=[17]-=-. 1.4 Involved Time Scales Time scales seen in neurobiology are much slower than those seen in most CMOS applications. Burst length times on the order of 0.3–10.0 seconds are common while fast spiking...

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... NERUON 10 2.3 Equation Analysis The analysis of the HR equations and similar systems have been a topic of interest in research. With respect to the chaotic operation of circuits, references [32] and =-=[33]-=- give an excellent primer into their operation. The simplest known circuit to exhibit chaotic operation, Chua’s Circuit, is analyzed. Its construction and analysis is similar to that of implementation...

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...onal amplifiers, two multipliers and 13 resistors (see figure 3.2). As most multipliers are differential in operation, extra circuitry is needed to condition signals to and from the multiplier blocks =-=[57]-=-. This leads to a standard integrator design that uses a large number of transistors and area and is a poor choice for a vast array on CMOS for robotic operation. A potential solution is to use either...

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...ehavior is created by central pattern generators(CPG) in both vertebrates an invertebrates. CPGs coordinate motion for walking, swimming, breathing and other motor funcCHAPTER 1. INTRODUCTION 4 tions =-=[15]-=-. These patterns are created from networks of neurons and synapses and operate even in the absence of descending commands from the brain or feedback from the senses [14]. CPGs can either control a sin...

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... from the need of photoreceptors and toward the use of electro-chemical sensors. CHAPTER 3. HINDMARSH-ROSE ANALOG COMPUTERS 26 Output LED Characterization Organic LEDs printed with “E-Jet” technology =-=[46]-=- is Cyberplasm project’s current method of approach for activating the synthetic muscles. Earlier work was toward the use of commercially available surface mount LEDs. The ROHM SMLP12BC7T was the smal...

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...ly meaningful. Can accurately account for the many ionic currents across the CHAPTER 1. INTRODUCTION 3 cellular membrane. Consists of 4 differential equations (expanded to over 7 by later researchers =-=[12]-=-) and has many tunable coefficients [13]. 1.2 Synapses Communication and coordination between neurons is made possible by synapses. When a synapse connects two neurons together the postsynaptic neuron...

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...phic robots use designs seen in nature to allow for better movement over terrain and obstacles. Examples include a robot fish that can operate wireless in three dimensions and have life-like movement =-=[22]-=-. Another is a robot snake that can be used to transverse harsh obstacles such as a debris field to search for survivors [23]. Researchers at the Northeastern University Marine Science Center have cre...

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...tal neuron is the subject of this chapter. Another advantage to using digital neurons is the ease of analysis of large neural maps [49] and the ease of reconfiguration to test many different theories =-=[50]-=-. Large maps of neurons require updating synaptic weights many times to fine tune operation, which is done much easier in code than in a hardware implementation [51]. 4.2 Theory of Operation The discr...

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...ated for. Mimicking nature is seen as a way to make robots more robust to environmental changes and variations as animals have evolved to adapt perfectly to their environments over thousands of years =-=[2]-=-. Invertebrate nervous systems are simple, easily identifiable and accessible using in vitro analysis. This has allowed researchers to make a one-for-one correspondence between an identified neuron or...

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... process repeats [6]. It is limited in the variety of outputs it can produce, usually only to tonic spiking. Another variation of this model type is the self-resetting neuron described by Carver Mead =-=[7]-=- and a similar model in reference [8]. In both, a capacitor is charged until a threshold voltage is reached at which point an output spike is produced and the capacitor voltage is reset. Resonate and ...

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...trol a single muscle, limb or another CPG. For example a CPG controls a leg in an insect, another controller would then pass timing control parameters to each leg CPG to coordinate walking or running =-=[16]-=-. Insects contain the simplest networks and can be easily studied and adapted for robotic use [17]. 1.4 Involved Time Scales Time scales seen in neurobiology are much slower than those seen in most CM...

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...es and discharges sets the neuron’s overall output time constant. A 1 pF capacitor charged by a 10 nA current charges to 1 V in 0.0001 seconds, much faster than the desired spike frequency. Reference =-=[18]-=- creates a spiking neuron in CMOS at a realistic output frequency of 33 Hz. This was accomplished using subthreshold operation and by locating the state capacitor off-chip. This allows a larger capaci...

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...in standard CMOS process. Many proposed CPG models simply operate too fast to control an electromechanical system such as a micro-robot. For example, a CPG circuit for rhythmic chewing is proposed in =-=[19]-=-. The time between chewing is shown to be 0.2 µS, which is orders of magnitude faster than its biological equivalent due in part to the sizing of the 60 pF capacitor. Reference [20] proposes a neuron ...

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...mselves were constructed to be as close as to nature as possible. Artificial muscles were created with nitinol wires which can be controlled similarly to the way live muscles are in natural organisms =-=[25]-=-. The above mentioned examples all share a similar control method: a high level finite state machine sending commands to various CPGs for locomotion. They are all running approximations of neural dyna...

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...ow the bursting output seen in mollusks [28]. Discrete electronic versions have also been compared to those seen in the crustacean stomatogastric system where a high degree of similarity was observed =-=[29]-=-. Electronic versions have even replaced neurons removed from a central pattern generator of a live lobster and have restored functionality [30]. 2.2 Three Coupled Differential Equations The three cou...

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...ARSH-ROSE NERUON 10 2.3 Equation Analysis The analysis of the HR equations and similar systems have been a topic of interest in research. With respect to the chaotic operation of circuits, references =-=[32]-=- and [33] give an excellent primer into their operation. The simplest known circuit to exhibit chaotic operation, Chua’s Circuit, is analyzed. Its construction and analysis is similar to that of imple...

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...everal values of I. Increasing I lowers the x nullcline causing the stable points to merge together then vanish. This drives the system into an unstable region and limit cycle, causing output spiking =-=[35]-=-. y = 1 a ( c x3 − b x2 − I ) (2.4) y = e− f x2 (2.5) CHAPTER 2. HINDMARSH-ROSE NERUON 11 Figure 2.2: Graph of the nullclines for the x and y variables. The intersection points can be found by setting...

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...reated from networks of summing circuits, integrators and trim pots for coefficient adjustments, they could be wired for a single purpose or rewired in a lab to solve a variety of simulation problems =-=[37]-=-. Such applications include process control, motor control, and electric grid control [38]. Differential equations have a one-for-one relationship in an analog computer. Reference [31] gives an overvi...

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...ments, they could be wired for a single purpose or rewired in a lab to solve a variety of simulation problems [37]. Such applications include process control, motor control, and electric grid control =-=[38]-=-. Differential equations have a one-for-one relationship in an analog computer. Reference [31] gives an overview of the implmentation in a discrete case but does not expand on scaling factors involved...

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... analog multiplier from Analog Devices were used in the design. The TL082 is a mid-end op-amp with low input bias current (typically 20 pA) due to its JFET input stage and a high slew rate of 16 V/µs =-=[42]-=-. Only a handful of companies still make analog multipliers as digital signal processing or application specific chips (such as variable filters) have become more widely accepted and cheaper. The AD63...

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...hs close to the outlined specification were to then be used as a controllable input. The requirement for photodiode inputs was the basis of a capacitive feedback transimpedance amplifier work done in =-=[45]-=- by another member of the team. Work was halted as the direction of the Cyberplasm project shifted away from the need of photoreceptors and toward the use of electro-chemical sensors. CHAPTER 3. HINDM...

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...e measured relationship between LED current consumption and forward voltage. Even with the switch to organic LEDs, a boost converter will be a future research topic due to their high turn-on voltages =-=[48]-=-. (a) Figure adapted from [44]. (b) Measured performance from test bench. Figure 3.10: Comparasion of measured LED performance with datasheet values. 3.8 Limitations A CMOS analog computer implementat...

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...on to test many different theories [50]. Large maps of neurons require updating synaptic weights many times to fine tune operation, which is done much easier in code than in a hardware implementation =-=[51]-=-. 4.2 Theory of Operation The discrete time–map based neural model is formed using a two-dimensional look-up function and a second equation which acts similarly to the z variable in the Hindmarsh-Rose...

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...o,n w l e Vgs−Vth,n nVT (5.2) 5.4.1 Schmitt Trigger The schmitt trigger is formed with five transistors. Four of them act as inverters with the fifth acting as a current source when the output is low =-=[59]-=-. This current source (transistor M5) is the reason for the hysteresis of the overall circuit. It increases the current through M2 when active thus transistor M1 must conduct more in order for the inv...