Author: logancollins

The simplest type of neural mass model involves assuming a homogenous population of neurons. This means that all neurons are coupled to each other and to themselves with an equal interaction strength w_ij=w₀. In graph theoretic terms, this is a complete graph with self-edges at every vertex. Furthermore, all neurons receive the same amount of externally applied current I^ext(t). As a consequence of these approximations, this type of model can only be used for large populations of neurons.
Neural coupling strengths that are less than zero are inhibitory. Neural coupling strengths that are greater than zero are excitatory.
Population activity is defined by the equation below. Note that this equation is not specific to homogenous populations, it can be used for many other types of models as well. N refers to the total number of neurons in the population while n_spikes counts the number of spikes between time t and a subsequent time t+Δt. The δ is the Dirac delta function and t_j^f is the time at which neuron j fires.

The electrophysiological activities of integrate-and-fire neurons are defined by the following differential equation and its solutions where τ_m is the membrane time constant (which equals the membrane resistance R times the membrane capacitance C), I(t) is the input current, and u is the membrane voltage. If the value of u passes a threshold ϴ, then a spike occurs and u is reset to the resting potential u_rest.

One of the tools necessary for describing the activity of a homogenous population of integrate-and-fire neurons is a function α(t – t_i^f) which represents the postsynaptic current generated by an input spike. Depending on the shape of the curve used to model the postsynaptic current, α(t – t_i^f) might take on different forms.
With all neurons are coupled to each other (and to themselves) in a homogenous population, the total current in any given neuron is the externally applied current plus the sum of all postsynaptic currents from input spikes multiplied by each interaction weight w_ij.

For homogenous populations with homogenous all-to-all coupling and a constant interaction strength w₀, the total current is the same in every neuron. This current is given by the following equation (since we can assume a continuum for a large population of neurons). The reason that the integral is multiplied by w₀N is that every neuron is connected to the given neuron. Here, s represents the time at which a spike occurs.

Consider a population in which each neuron has slightly different parameters from the others such that the firing rates r_i(I(t)) vary over the population despite each neuron receiving the same input current. If the population is large, then the function which describes the variation in firing rate can be linearized around the average firing rate (and neglecting the higher-order terms of the Taylor series). As such, this simplification (the linearized model) can still be useful for some applications.

The above expression can also be thought of as indicating that the mean firing rate of the population is equal to the firing rate of a “typical” neuron (with “typical” parameters) in the population.
In cases that involve more dramatic variations within populations of neurons, the averaging technique described above is insufficient. For instance, consider a population in which half the neurons are described by a set of parameters p₁ and the other half by a set of significantly different parameters p₂. This population should be split and regarded as two homogenous populations.
Indeed, any population composed of subsets which differ significantly from each other should be decomposed into the homogenous subsets. The same applies to populations composed of neurons with identical parameters, but with subsets that receive significantly different input currents.

Connectivity schemes and scaling

Using these techniques, populations of different sizes can give similar results if a scaling law is applied to the connection weights. For a homogenous population with all-to-all connectivity, the appropriate scaling law is as follows. J₀ is the number of neurons before scaling and N is the number of neurons after scaling.

Increasing the size of a population while keeping its connectivity the same allows for noise reduction. This is especially useful since some populations are quite small. For instance, a single layer within a cortical column might have only a few hundred neurons.
Another all-to-all coupling model involves using a Gaussian distribution of weights with the following mean and standard deviation (σ₀ is the standard deviation of weights prior to scaling).

Populations can also be modeled by setting a fixed coupling probability p (among N² possible connections). In this type of model, the mean number of connections to a neuron j is then given by pN and the variance is p(1 – p)N. Alternatively, each neuron j can send outputs to pN partners. To scale a population with a fixed coupling probability, the equation below is used so that the average number of inputs to each neuron does not change as the population size changes.

Some simulations can assume a balanced population of excitatory and inhibitory neurons. In such cases, the mean input current is zero, so scaling the connection weights does not influence the mean. Instead, the weights should be scaled with respect to how they affect fluctuations about zero. This can be achieved using the following scaling equation.

Interacting populations

In the previous sections, balanced populations of excitatory and inhibitory neurons were used. Now, consider homogenous populations each consisting of either excitatory or inhibitory neurons, but not both.
These populations can be visualized as spatially separated from each other, but this is not necessary for the model to work (and it is not biologically realistic). The populations could just as easily be spatially mixed.
The activity of neurons in homogenous population n is given by the equation below. The parameter Γ_n represents the set of neurons belonging to population n.

With all-to-all coupling, each neuron i within pool n is assumed to receive inputs from every neuron j within pool m. The connection strength is w_ij=J_nmN_m where J_nm is the strength of an individual coupling from a neuron in pool m to a neuron in pool n and N_m is the number of neurons in pool m. As such, the input current to a neuron i will come from all the spikes in the network. Once again, α represents some given type of postsynaptic voltage-time function after an input current.

The input current can also be formulated by the equation below. Since the model provides identical input current to all neurons, the index i can be removed. Once again, s represents the time at which a spike occurs.

Distance-dependent connectivity

To better model neural populations, distance can provide an approximate measure for coupling probability (with more distant neurons having a lower probability of coupling). It should be noted that this is still a very rough model.
In order to create a model with distance-dependent connectivity, each neuron i must be assigned a location x(i) on a two-dimensional cortical sheet.
For this type of model, all connections are assigned the same weight and the connection probability depends on distance (see part A of the diagram). P is a function which maps any vector to a real number on the interval [0,1].

Alternatively, all-to-all coupling can be assumed with a strength w_ij that decreases with distance (see part B of the diagram). This is modeled by the following equation where g is a function that maps any vector to a real number. (For example, a Euclidean length metric inside of a decaying exponential).

Spatial continuum models

Many neural populations in the brain exhibit properties which continuously vary across space (i.e. tonotopy and retinotopy). Of course, this kind of variation is not actually continuous at the level of individual neurons, but it is effectively continuous from the perspective of population modeling. Sets of homogenous populations cannot account for such continuous variation, so spatial continuum models must be used when considering this kind of functional organization.
Consider a continuum of neurons along a one-dimensional axis and assume all-to-all coupling with connection strength dependent upon distance. This model uses the equation w_ij = g(|x(i) – x(j)|) as described in the previous section. Then discretize space into segments of length d. The number of neurons in a segment n is given by the following equation where ρ represents the density of neurons. Neurons within this interval belong to the set Γ_m.

For continuum models, the population activity of population m is described as a function of time and of the spatial position of the neurons belonging to population m. The latter is given by md since the distance along the axis is equal to the segment length times the index m.
The coupling strength between a neuron x(i) at location nd and a neuron x(j) at location md is a function w(nd,md) that defines a weighting measure depending on the distance between the two locations nd and md.
As such, the input current to a neuron in population m is computed using the following equation (top). The product of an input current from population n with the population activity induced in population m is found inside of the summation. All input currents to population m from the rest of the populations are then summed. For a large number of populations, this equation can be replaced by an equation with a second integral. For convenience, md has been replaced with y in these equations.

Reference: Gerstner, W., Kistler, W. M., Naud, R., & Paninski, L. (2014). Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition. New York, NY, USA: Cambridge University Press.

Notes on – Observing the cell in its native state: imaging subcellular dynamics in multicellular organisms (Betzig et al.)

September 7, 2018May 31, 2019By logancollins No Comments

PDF Version: Notes on – Observing the cell in its native state imaging subcellular dynamics in multicellular organisms (Betzig et al.) – Logan Thrasher Collins

Overview

Eric Betzig, a 2014 Nobel laureate in chemistry (for playing a key role in developing super-resolution fluorescence microscopy), has continued to advance biological imaging technologies.
In their 2018 paper, “Observing the cell in its native state: imaging subcellular dynamics in multicellular organisms,” Betzig’s group combined lattice light sheet microscopy and adaptive optics to acquire 3-dimensional images and videos of physiological processes in vivo with subcellular resolution, achieving unprecedented detail and clarity.

Image source: (Liu et al., 2018)

Light sheet microscopy

Light sheet microscopy uses a laser and a cylindrical lens to project a plane of illumination through a sample.
As the sheet only illuminates a single plane of the sample at a time, this technique decreases photodamage relative to other methods which pass light through the entire sample at once. In addition, the single plane illumination decreases background noise and so generates images with high contrast.
Light sheet microscopy operates rapidly since it scans entire layers of the sample all at once rather than scanning one point of light at a time (the latter is common with other types of microscopy).
Each layer of the sample is imaged in this way before the layers are stacked to reconstruct a 3-dimensional image. Betzig’s group used a mathematical algorithm called deconvolution to remove out-of-focus light and enhance more focused light within the Z-stacks generated by his lattice light sheet microscopy technique.

Image source: Jan Krieger, CC BY-SA 3.0, commons.wikimedia.org

Lattice light sheet microscopy

In lattice light sheet microscopy, the input laser is first stretched (in the x direction) by a pair of cylindrical lenses and then compressed (in the z direction) into a sheet by another pair of cylindrical lenses oriented perpendicularly to the first pair.
Lattice light sheet microscopes combine Bessel beams with 2D optical lattices using an optical element called a spatial light modulator. The spatial light modulator uses a ferroelectric liquid crystal display to create programmable gratings that diffract incoming light into a customized pattern. This is displayed in the figure from Förster et al., where a spatial light modulator displays a customized grid of white squares (light can pass through) and black squares (light is blocked) which forms a diffraction grating.
- Bessel beams are fields of electromagnetic radiation which can be mathematically described by Bessel functions of the first kind. Unlike most electromagnetic radiation, Bessel beams do not spread out as they propagate. As such, they do not form diffraction patterns. Although ideal Bessel beams are physically impossible, approximate forms of the phenomenon can be created.
- 2D optical lattices arise from the interference of beams of light that exhibit periodic behavior in two dimensions. They can form the same set of patterns found in 2D Bravais lattices (which are a mathematical formulation for crystal structures).
On their own, neither Bessel beams nor 2D optical lattices are useful for light sheet microscopy, but they achieve superior properties for imaging when properly implemented together.
- Bessel beams contain the same amount of energy in their “side lobes” as they do in their central peaks. For this region, they cause excessive illumination outside of the targeted plane. This is problematic for light sheet microscopy since the technique depends on having a focused plane of light.
- Despite their name, 2D optical lattices extend into 3D space (since many “copies” of the planar lattice occur along the z direction). This is similarly problematic for light sheet microscopy since the technique depends on the light exhibiting confinement to the xy plane of focus.
- These issues can be overcome by combining Bessel beams and 2D optical lattices. To achieve this, a ring (or annulus) of illumination is used to destructively interfere with the side lobes of the Bessel beams, creating Bessel-Gauss beams. Then an array of coherent Bessel-Gauss beams is generated (two waves with a constant phase difference, the same frequency, and the same waveform are coherent). This array of Bessel-Gauss beams is suitable as a light sheet for lattice light sheet microscopy.

Image source: (Förster et al., 2014)

Adaptive optics

The light used for excitation in lattice light sheet microscopy traverses different regions of the sample relative to the detected light. As such, the light involved in excitation and the light involved in detection are subject to different aberrations.
To adjust for such aberrations, Betzig’s group used a two-photon excitation beam from an ultrafast Ti:Sapphire laser. This beam creates a “guide star” (this term comes from a similar technique used in astronomy) which acts as a reference. The guide star is scanned over the entire focal plane so as to compute an average correction since average correction has more accuracy than a correction from a single point in the sample.
Next, a switching galvanometer SG1 (a device which rotates a mirror back and forth) facilitates transfer of the two-photon excitation beam to either the excitation or detection arms of the microscope as needed.
For the detection beam, the light generated from the scanned guide star is collected and sent to a device called a Shack-Hartmann wavefront sensor using another switching galvanometer SG2.
- The Shack-Hartmann wavefront sensor contains an array of small sensors which measure the “tilts” of the incoming plane waves. By measuring the local tilt of each small wavefront composing the detected light beam, the overall shape of any optical aberration can be approximated.
- Then a deformable mirror is modified to precisely compensate for the aberration measured by the Shack-Hartmann wavefront sensor. The second switching galvanometer SG2 transfers the light to this deformable mirror. After reflecting from the deformable mirror, the aberration is corrected and the detection objective collects the light to create an image.
For the excitation beam, the light from the two-photon excitation laser is scanned over the sample as a guide star and then collected by the excitation objective.
- The collected light is transferred to another Shack-Hartmann wavefront sensor. Once again, the sensor measures an approximate representation of any optical aberration which occurs.
- Next, the spatial light modulator used in creating the light sheet itself applies the appropriate correction to the excitation beam. This is highly effective since the spatial light modulator provides exquisite control over the output light sheet.

Image source: (Liu et al., 2018)

References

Chen, B.-C., Legant, W. R., Wang, K., Shao, L., Milkie, D. E., Davidson, M. W., … Betzig, E. (2014). Lattice light-sheet microscopy: Imaging molecules to embryos at high spatiotemporal resolution. Science, 346(6208). Retrieved from http://science.sciencemag.org/content/346/6208/1257998.abstract

Förster, R., Lu-Walther, H.-W., Jost, A., Kielhorn, M., Wicker, K., & Heintzmann, R. (2014). Simple structured illumination microscope setup with high acquisition speed by using a spatial light modulator. Optics Express, 22(17), 20663–20677. http://doi.org/10.1364/OE.22.020663

Liu, T.-L., Upadhyayula, S., Milkie, D. E., Singh, V., Wang, K., Swinburne, I. A., … Betzig, E. (2018). Observing the cell in its native state: Imaging subcellular dynamics in multicellular organisms. Science, 360(6386). Retrieved from http://science.sciencemag.org/content/360/6386/eaaq1392.abstract

Kosmik Kaleidoscope (poem)

August 29, 2018August 29, 2018By logancollins No Comments

This poem was featured on the blog run by Paul Anlee, author of the Deplosion series. Check out the post at www.PaulAnlee.com

Kosmik Kaleidoscope

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