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Bioengineering applies engineering methods to biological systems. A physiological model represents a system with equations or circuits so that behaviour can be predicted and tested. Levels: molecular, cellular, tissue, organ and system. Good models are as simple as possible while capturing the phenomenon, with parameters that can be estimated from measurements.

Ion movement across the membrane is driven by concentration gradient (diffusion) and electric field (drift), described by the Nernst-Planck equation. At equilibrium for a single ion the Nernst equation gives the potential: E = (RT/zF) ln([ion]out/[ion]in), about 61 mV x log10(ratio)/z at body temperature. Typical values: E_K about -90 mV, E_Na about +60 mV. With several permeant ions the Goldman-Hodgkin-Katz equation weights each by its permeability: the membrane potential lies near the equilibrium potential of the ion with highest permeability (potassium at rest, about -70 mV). Parallel conductance model: the membrane is a capacitor (about 1 microfarad per cm^2) in parallel with a conductance and battery for each ion: Vm = sum(g_i E_i)/sum(g_i).

Hodgkin and Huxley (squid giant axon) described the membrane current as capacitive current plus sodium, potassium and leak currents: C dV/dt = -(I_Na + I_K + I_leak) + I_applied, with conductances g_Na and g_K controlled by gating variables m, h (sodium activation and inactivation) and n (potassium activation), each following first-order kinetics with voltage-dependent rate constants. Sequence: depolarisation above threshold opens sodium channels (positive feedback, upstroke), then sodium inactivation and delayed potassium opening repolarise the cell, followed by a refractory period. The model reproduces threshold, all-or-none behaviour and propagation.

Cable model: the axon has axial (internal) resistance and a membrane with resistance and capacitance per unit length. Length constant lambda = sqrt(r_m/r_i): the distance over which a steady voltage decays to 37 percent; larger for large diameter and high membrane resistance. Time constant tau = r_m c_m (or R_m C_m): the time for the membrane to charge to 63 percent. Conduction velocity increases with fibre diameter (proportional to the square root of diameter in unmyelinated fibres). Myelination raises membrane resistance and lowers capacitance, so the signal jumps between nodes of Ranvier (saltatory conduction) and velocity is much higher, roughly proportional to diameter.

1. State the Nernst equation. 2. Why is the resting potential close to E_K? 3. Name the gating variables in the Hodgkin-Huxley model. 4. Define the length constant. 5. Why is conduction faster in myelinated fibres?

1. E = (RT/zF) ln([ion]out/[ion]in). 2. At rest the membrane is most permeable to potassium. 3. m and h for sodium, n for potassium. 4. The distance over which a steady voltage decays to about 37 percent, lambda = sqrt(r_m/r_i). 5. Myelin increases membrane resistance and decreases capacitance, allowing saltatory conduction between nodes.

The heart is modelled as a time-varying current dipole (the cardiac vector) in a volume conductor. Depolarisation towards a positive electrode gives a positive deflection. Einthoven's triangle: electrodes on the right arm (RA), left arm (LA) and left leg (LL) define bipolar leads I (LA - RA), II (LL - RA) and III (LL - LA), with Einthoven's law: II = I + III. Goldberger's augmented leads (aVR, aVL, aVF) measure each limb electrode against the average of the other two. The precordial leads V1-V6 are unipolar leads on the chest with the Wilson central terminal as reference. Together they give a 12-lead ECG viewing the heart in frontal and horizontal planes.

Signal: amplitude about 1 mV, bandwidth about 0.05-100 Hz (diagnostic) with 150 Hz for paediatric use. Requirements: differential amplifier with high input impedance and high common-mode rejection ratio (CMRR), to reject 50 Hz power-line interference which is common to both electrodes; right-leg drive; Ag/AgCl electrodes with gel to reduce contact impedance; filtering for baseline wander (high-pass) and muscle noise; galvanic isolation for patient safety; then sampling (at least 250 Hz) and digitisation.

1. State Einthoven's law. 2. How are the augmented limb leads defined? 3. Why is a high CMRR needed in ECG amplifiers?

1. Lead II = lead I + lead III. 2. Each limb electrode is measured against the average of the other two. 3. To reject 50 Hz interference and other signals common to both electrodes.

Fourier series and transform: a periodic signal is the sum of harmonics; the Fourier transform gives the spectrum (amplitude and phase versus frequency) of a general signal. Sampling: the signal is measured at intervals T. Nyquist theorem: the sampling frequency must exceed twice the highest frequency (bandwidth) present; otherwise aliasing occurs, so an analogue anti-aliasing low-pass filter precedes the converter. Quantisation: an n-bit converter has 2^n levels; the quantisation step is the full-scale range divided by 2^n, and the quantisation noise reduces the signal-to-noise ratio (about 6 dB per bit).

The discrete Fourier transform (DFT) computes the spectrum of N samples; the fast Fourier transform (FFT) computes it in about N log N operations. Frequency resolution is the sampling rate divided by N. Filters: low-pass, high-pass, band-pass and band-stop (notch, to remove 50 Hz). Digital filters can be FIR (finite impulse response, always stable, linear phase) or IIR (infinite impulse response, more efficient, may be unstable). Filter design specifies cut-off frequency, order and ripple.

1. State the Nyquist theorem. 2. What does an anti-aliasing filter do? 3. Difference between FIR and IIR filters? 4. How does the number of bits affect the signal-to-noise ratio?

1. The sampling rate must be more than twice the highest signal frequency. 2. Removes frequencies above half the sampling rate before conversion. 3. FIR: no feedback, always stable, linear phase; IIR: feedback, more efficient, can be unstable. 4. About 6 dB improvement per additional bit.

Recording: electrodes placed by the international 10-20 system (spacing at 10 and 20 percent of skull distances; odd numbers on the left, even on the right; letters F, C, P, T, O for regions). Signals are microvolts in amplitude, so a differential amplifier with high gain and CMRR is needed. Montages can be referential or bipolar. Rhythms: delta (below 4 Hz, deep sleep), theta (4-8 Hz, drowsiness), alpha (8-13 Hz, relaxed with eyes closed, occipital), beta (13-30 Hz, active thinking) and gamma (above 30 Hz). Applications: epilepsy, sleep staging, brain-computer interfaces. Artefacts: eye blinks, muscle activity and electrode motion.

Electrodermal activity (EDA) is the change in skin conductance caused by sweat secretion. Components: slow tonic level (skin conductance level) and fast phasic responses (skin conductance responses, peaking one to three seconds after a stimulus). Measured with two electrodes on the fingers or palm and a small constant voltage. Used in emotion recognition, stress monitoring and lie detection research, often combined with heart rate variability.

1. What does the 10-20 system define? 2. Which EEG rhythm is typical of relaxed wakefulness with eyes closed? 3. Why does electrodermal activity reflect sympathetic arousal?

1. The standardised positions of scalp electrodes at 10 and 20 percent of skull distances. 2. Alpha rhythm (8-13 Hz). 3. Sweat glands are innervated only by the sympathetic system.

A control system has a controlled variable, sensor, controller and actuator. In the baroreflex, arterial pressure is the controlled variable, baroreceptors the sensor, the brainstem the controller and heart rate and vascular tone the actuators. The Frank-Starling mechanism is a local mechanical control of stroke volume. Guyton's circulatory model represents the closed circulation with venous return, cardiac output and vascular resistance curves; the intersection of the cardiac output curve and the venous return curve gives the operating point. Long-term control of pressure depends on the kidney (fluid balance).

Sensory feedback (vision, vestibular, proprioception) and delays in the neural loop make motor control a feedback problem with time delay. In the inverted pendulum model, the body sways around the ankle; gravity destabilises it, and the calf muscles apply a corrective torque. EMG measures the electrical activity of muscle: the surface EMG is an interference pattern of motor unit action potentials, amplitude 0.1-5 mV, bandwidth 20-500 Hz; the envelope of the rectified signal estimates muscle activation.

1. Identify sensor, controller and actuator in the baroreflex. 2. What does the intersection of cardiac output and venous return curves represent? 3. Why is quiet standing modelled as an inverted pendulum? 4. Which frequency band does the surface EMG occupy?

1. Sensor: baroreceptors; controller: brainstem; actuators: heart and vessels. 2. The operating point of the circulation. 3. The body's centre of mass is above the ankle, which is an unstable configuration stabilised by muscle torque. 4. About 20-500 Hz.