Key Formulas & Equations
Core Concepts
Neuron Structure
The fundamental unit of the nervous system. Three parts: dendrites (receive signals), soma/cell body (integrate signals), axon (transmits signals). Myelin sheath insulates axons, enabling saltatory conduction (50-120 m/s). Human brain contains ~86 billion neurons.
Action Potential
Electrical impulse that travels along a neuron. At rest: ~-70 mV. Depolarization: Na⁺ channels open, membrane rushes toward +40 mV. Repolarization: K⁺ channels open, membrane returns to rest. Refractory period: ~1-2 ms when neuron cannot fire again. All-or-none principle.
Synaptic Transmission
Communication between neurons via neurotransmitters. Action potential reaches terminal → Ca²⁺ influx → vesicles release neurotransmitter → binds receptors on postsynaptic neuron. Types: excitatory (EPSP, depolarizing) and inhibitory (IPSP, hyperpolarizing). Major neurotransmitters: glutamate (excitatory), GABA (inhibitory), dopamine, serotonin, acetylcholine.
Neuroplasticity
The brain's ability to reorganize by forming new synaptic connections throughout life. Mechanisms: long-term potentiation (LTP — strengthening synapses with repeated use) and long-term depression (LTD — weakening unused synapses). Basis for learning and memory. Enables recovery from brain injury.
Brain Anatomy
Major regions: Brainstem (breathing, heart rate), Cerebellum (motor coordination), Hypothalamus (homeostasis, hormones), Hippocampus (memory formation), Amygdala (emotions, especially fear), Frontal lobe (decision-making, planning), Parietal lobe (sensory integration), Temporal lobe (auditory processing, language), Occipital lobe (vision).
Neuroimaging
Techniques to visualize brain structure and function. MRI: detailed anatomical images using magnetic fields. fMRI: measures brain activity via blood flow (BOLD signal). EEG: records electrical activity from scalp (excellent temporal resolution). PET: uses radioactive tracers to show metabolic activity. CT: X-ray-based structural imaging.
Worked Examples
Example 1: Nernst Equation
Problem: Calculate E_K for a neuron with [K]_in = 140 mM and [K]_out = 5 mM at 37°C. (R=8.314, F=96485, z=+1)
Step 1: E_K = (RT/zF) × ln([K]_out/[K]_in)
Step 2: E_K = (8.314 × 310)/(1 × 96485) × ln(5/140)
Step 3: E_K = 0.0267 × ln(0.0357) = 0.0267 × (-3.33)
Step 4: E_K ≈ -89 mV
Answer: The potassium equilibrium potential is approximately -89 mV.
Example 2: Receptor Binding
Problem: A neurotransmitter with K_d = 10 μM binds to a receptor. What fraction of receptors are occupied at [L] = 30 μM?
Step 1: Fractional occupancy = [L]/(K_d + [L])
Step 2: f = 30/(10 + 30) = 30/40
Step 3: f = 0.75 = 75%
Answer: 75% of receptors are occupied at 30 μM ligand concentration.
Example 3: Time Constant
Problem: A neuron has R_m = 10 MΩ and C_m = 100 pF. What is the membrane time constant?
Step 1: τ = R_m × C_m = 10×10⁶ × 100×10⁻¹²
Step 2: τ = 10⁷ × 10⁻¹⁰ = 10⁻³ s
Step 3: τ = 1 ms
Answer: The membrane time constant is 1 millisecond.
Example 4: Fick's Diffusion
Problem: A neurotransmitter diffuses with D = 3×10⁻⁶ cm²/s. If concentration drops from 1 mM to 0.1 mM over 0.01 cm, find flux J.
Step 1: dC/dx = (0.1 - 1.0)/0.01 = -90 mM/cm
Step 2: J = -D × dC/dx = -(3×10⁻⁶) × (-90)
Step 3: J = 2.7×10⁻⁴ mM/(cm²·s) = 270 nM/(cm²·s)
Answer: The flux is 270 nM/(cm²·s) toward the lower concentration region.
Practice Problems
Problem 1
What is the primary excitatory neurotransmitter in the mammalian CNS?
Answer: Glutamate (acts on NMDA and AMPA receptors)
Problem 2
Calculate the Nernst potential for Na⁺ with [Na]_in = 15 mM, [Na]_out = 145 mM at 37°C. (z = +1)
Answer: E_Na = 0.0267 × ln(145/15) = 0.0267 × 2.27 ≈ +60 mV
Problem 3
What EEG frequency band dominates during deep sleep (N3 stage)?
Answer: Delta waves (0.5-4 Hz)
Problem 4
What is the typical resting membrane potential of a neuron?
Answer: Approximately -70 mV
Problem 5
During an action potential, which ion rushes IN first?
Answer: Sodium (Na⁺) through voltage-gated Na⁺ channels
Problem 6
Hebbian learning rule: what happens when two neurons fire together repeatedly?
Answer: "Neurons that fire together wire together" — synaptic connection strengthens (LTP)