What gets measured, what gets modeled, and how they connect

Before any of the formal quantities — transfer functions, impedance, coherence, normal modes — have any meaning, you need to know what a neuroscientist actually measures in a lab and what derived numbers they care about. This page walks from the raw observables up through the secondary statistics, then crosses over to the quantities a model predicts. By the time we get to $H(\omega )$, you'll see that it's the model-side counterpart of the most-recorded thing in human neuroscience: a power spectrum.

What a neuroscience experiment produces

Three kinds of raw data dominate, each at a different spatial scale:

1. Single-neuron voltage trace V(t)

A glass micropipette pushed through the membrane of one neuron (intracellular recording, patch clamp). You see the subthreshold voltage fluctuations and the action potentials it produces. Sampled at 10-50 kHz, picks up everything from synaptic noise (~mV) to spikes (~100 mV). The cleanest signal you can get from a brain, but you only get one neuron at a time and it usually dies within an hour.

2. Population electrical activity

A weighted sum of many nearby neurons' currents, measured at progressively wider spatial scales:

• Local Field Potential (LFP) — extracellular electrode inside the brain, picks up the summed synaptic currents (mostly excitatory inputs onto pyramidal cells) of neurons within roughly 100-300 μm.
• Multi-electrode array (MEA) — a grid of electrodes, each LFP-ish, sampled simultaneously.
• Electrocorticography (ECoG) — electrodes on the cortical surface; coarser spatial resolution, used in clinical epilepsy and BMI work.
• Electroencephalography (EEG) — scalp electrodes; sums currents from hundreds of thousands of neurons through the skull. Cheap and non-invasive, but the spatial resolution is poor and high frequencies are filtered out.
• Magnetoencephalography (MEG) — SQUID magnetometer outside the head; complementary to EEG (sees tangential currents that EEG misses).

All of these give you a time series of voltages (or magnetic fields). The signal at any one moment is a sum of many neurons' contributions; the spatial scale determines how many neurons are summed.

3. Spike trains

From $V(t)$ or a multi-unit recording, threshold-detect the action potentials and keep only the spike times $\{t_1,t_2,\dots \}$. A point process — discrete events with no amplitude information. This is what most information-theoretic and rate-based modeling cares about.

Secondary quantities derived from the raw recording

Given a time series $x(t)$ or a spike train, what do you actually compute? Six standard things:

Mean firing rate ν

Spikes per second. Baseline summary. For a single neuron, often 1-50 Hz in awake cortex.

Autocorrelation R(τ)

How does the signal correlate with itself, delayed by $\tau$? Fast decay means broadband, slow decay means long memory or oscillation. The width of $R(\tau )$ near zero is the signal's intrinsic timescale.

Power spectrum S(ω)

The Fourier transform of the autocorrelation, equivalently the squared modulus of the signal's Fourier transform. This is what gives you headline neuroscience phrases like "alpha rhythm at 10 Hz" or "gamma at 40 Hz" — peaks in $S(\omega )$ at those frequencies. Most human EEG/MEG studies that report a "spectrum" mean exactly this.

Coherence C_ij(ω)

For two simultaneously-recorded signals (e.g., LFP at two depths, or EEG at two scalp locations), how synchronized are they at each frequency? $C\in [0,1]$, with 1 meaning perfectly coherent. This is the standard tool for studying brain-rhythm communication between regions.

Tuning curve

Firing rate (or response amplitude) as a function of a stimulus parameter. Visual neurons in V1 have orientation tuning curves; auditory neurons in A1 have frequency tuning curves; motor neurons have direction tuning curves. The width and peak of the tuning curve are the neuron's "preferred" stimulus and its selectivity.

Spike-triggered average (STA)

The average stimulus pattern over a window just before a spike. Tells you what input feature drives the neuron. For visual neurons it gives a spatial receptive field; for auditory neurons, a spectro-temporal receptive field; in any setting, it's the most basic estimate of the input-output filter.

What a model predicts

Now flip to the model side. You have a network: connectivity $W$, time constants, neuron model. What can you compute about it?

Mean firing rates ν_i

From a self-consistency equation (Brunel 2000, Wilson-Cowan). For each neuron, given the population input it receives, predict its steady-state firing rate. Solving for self-consistency gives the population's mean rates in the asynchronous state. Directly comparable to recorded mean firing rates from experiments.

Transfer function H(ω)

The response amplitude at neuron $j$ when neuron $i$ is driven by an oscillating current at frequency $\omega$. $L$ is the linear-response operator of the network. This is the quantity the LIF network response page computes via Lanczos+CF.

Network eigenvalues

The eigenvalues of $L$ are the network's natural frequencies and damping rates. They're also the poles of $H(\omega )$. From an experimentalist's view: these are the network's intrinsic rhythms (real part of the eigenvalue = damping rate, imaginary part = oscillation frequency).

Predicted power spectrum

For a model driven by noisy input $\eta (t)$ with white noise power $P_{\eta }$, the predicted power spectrum at neuron $i$ is

This is the formula that lets you predict what an LFP recording will look like, from a network model. For more sophisticated noise (correlated across neurons, frequency-dependent, etc.) this generalizes, but the structural relationship spectrum is squared modulus of the transfer function holds in linear response.

Predicted coherence

The cross-region coherence is built from off-diagonal entries of the transfer-function matrix. So an experimentally-measured coherence between two brain regions reflects the off-diagonal response structure of the underlying network.

The bridge: measurement ↔ model

Row by row, what the experimentalist measures and what the modeler computes:

Each row is a predict-and-compare cycle: compute the model quantity, look at the experimental quantity, compare. Match means the model captured something real; mismatch means something's wrong — connectivity statistics, time constants, excitatory-inhibitory balance, choice of neuron model.

So what is H(ω), in one sentence?

$H(\omega )$ is the model-side counterpart of the most common quantity neuroscience experiments produce — a power spectrum — together with information about phase and cross-region propagation that a pure power spectrum doesn't carry. Computing $H(\omega )$ from a network model is what lets you predict the measurable spectrum, the measurable coherence, and the measurable rhythm peaks, all at once, from a single resolvent-matrix-element calculation.

Which method for which quantity

Computational neuroscience has a wide method zoo because brains span an enormous range of scales — from one ion channel to a whole connectome. No single method covers everything. Pick the method that matches the quantity you need and the network size you're working with:

The methods are roughly ordered top to bottom by spatial scale: biophysical single-cell at the top, whole-brain inference at the bottom, with the linearized-response and mean-field methods bridging them. Lanczos+CF sits in that bridge — the right tool when you have a sparse linear operator on a giant network and want its frequency-domain response.

Related on this site

• Leaky integrate-and-fire — the single-neuron model that, networked together, produces the $H(\omega )$ we discuss here.
• LIF network linear response via Lanczos+CF — the computation, the convergence analysis, and the connection to circuit MOR.
• Lanczos MOR for large circuits — the same machinery on VLSI interconnect networks, with explicit derivation of the Padé-via-Lanczos correspondence.
• Lanczos iteration — the underlying algorithm and the tridiagonal-inverse-is-a-continued-fraction theorem.