Group velocity dispersion memo

Mode-lock pulsed laser has broad wavelength spectrum due to the Heisenberg principle:
Δν τ = τ cΔλ/(λ²)~ 0.44 (FWHM). For 100fs, the wavelength width (Δλ) is ~8nm at 800nm wavelength.

Because light with different wavelength propagates with different velocity, the dispersion of pulsewidth occurs.

Phase shift of light in some material is:

Φ (ω) = n(ω)Lω/c (1)

where L is the length of material, n is the refractive index and c is the speed of light. Φ(ω) can be described as a power seriese as:

Φ(ω) = Φ(ω₀) + (dΦ(ω)/dω) *(ω-ω₀) + (d²Φ(ω)/dω²) * (ω-ω₀)²

The first and seond term does not change the pulse shape. The third term causes pulse broadening. The second order differentiation of Φ can be obtained from equation (1) as:

(d²Φ(ω)/dω²)L^-1 = λ³(2πc²) ^-1 (d²n/dλ²)

The left term is refered to as group velocity dispersion constant, D. Typical value is 100-400 fs^2/cm for silica or BK7 and 1000-2000 fs^2/cm for SF10 or SF11. Pulse broadening is expressed as:

τ_out = τ_in[1 + 7.68*(DL)² τ_in^-4]^1/2
=τ_in[1+205*(DL)² (cΔλ / λ²)⁴]^1/2

Here 7.68 = (2*sqrt(ln2))^4 is to covert gaussian width parameter to FWHM. DL of typical microscope setting is 3,000-10,000 fs². Compensation can be done using a pair of prism with distance of L_prism obtained by:

D_prism = -2L_prismλ³(πc²)^-1(dn/dλ)²

For example, high dispersive materials such as SF-10 or SF-11 have dn/dλ = 40,000-80,000 m^-1 at 800-900nm. Thus, L_prism~30-100cm is required to compensate the dispersion caused by a microscope.

---
Sellmeier equation

λ dependency of dispersive materials can be described as:
n²=1+Σ(A_iλ²)/(λ²-B_i)

A1,A2,A3,B1,B2,B3
SF10 1.61625977 0.25922933 1.07762317 0.0127535 0.05819840 116.607680
SF11 1.73848403 0.31116897 1.17490871 0.0136069 0.06159605 121.922711

References:
V.Iyer et al. J.Biomed.Optic 2006
(http://sensor.bcm.tmc.edu/saglab/pdf/JBO_Iyer.pdf)

Newport web page
(http://www.newport.com/store/genproduct.aspx?id=141161〈=1033&section=Detail
)

Mellesgriot web page
http://www.mellesgriot.com/products/optics/mp_3_1.htm

Wikipedia Sellmeier equation
http://en.wikipedia.org/wiki/Sellmeier_equation

BCP webpage
http://bcp.phys.strath.ac.uk/ultrafast/dictionary/dispersion%20and%20pulse%20broadening/dispersion%20and%20pulse%20broadening.html

Building a 2-photon microscope -- 2

PMT high voltage power-supply: Using Hamamatsu C4900-1 in combination with 12V powersuply (Power One) is a cost efficient solution. For controlling voltage, one needs a potentiometer (50 ohm).(~$0.2K each).

XYZ-motorized control: I use XY-stage from NEAT (XYR-4040), and Z-axis from Oriental motor (PK266-02A) coupled to microscope focus control. Cables are connected to Sutter MP285. Originally MP285 is for electrode manipulator, but one can ask for modification for the NEAT stage (KS3.4). $7K.

Building 2-photon microscope -- 1

As I am setting up a 2-photon microscope in my lab, I decided to leave my note here. The system should be significantly cheaper (say, 50%) and probably better than any commercial 2-photon systems, but obviously you need to commit more time and efforts to build.

Materials

- Microscope: I use olympus upright microscope, IX51, because their fluorescence microscope is featured with extremely low background light, so that one can even visualize single molecule without any special setting. I asked them to make a hole (~35 mm) in the side of a filter cube turret, and screw holes around the hole that fits to microconstruction system from Thorlabs. Also, I use a dichroic mirror holder that reflects light to the side (Olympus now provide this cube). This way, we can put photo-multiplier (PMT) to the side of microscope. It should be noted that we need to put PMT as close as possible to the objective to get maximum number of photons in light-scattering tissues. (~$40K)

- mirrors, lens, posts, holders: Thorlabs and Newport (~$1K)

- Optical table: I use big table (12 x 4 feet) that can hold two rigs. Because lasers are so expensive, it is cost effcient to split one laser light and feed into two rigs. It is a little bit tricky to do electrophysiology in two rigs in a table, but still doable. (~$8K)

- Scanner: Cambridge technologies provides fairly good and cheep scanner (6201H). Their lead time is ridiculously long (9 weeks). Their driver board get very hot during opperation. One should not put a lid on this. (~$2K)

- Scan lens: I purchased scan lens for a confocal microscope from Olympus. If one wants to reduce cost, it is good enough with a standard achromatic lens. Singlet lens could also work. (~$1K)

- Machining: One needs to machine an adaptor to connect scanner to the scan lens.

- Cage: One needs complete darkness for 2-photon unlike confocal system. I made a box (size = hight 36''x width 30''x depth 30'') made of angled and slotted bars (McMaster.com, 4664T51) and thin alminum plates (89015K71
). Inside of the cage is black-painted with a car painting spray. Make several holes for wiring and laser light. Metal cage also reduces electrical noise. (~$0.2K)

- Signal amplification and filtering: It is necessary to filter photon signal from PMT with ~300KHz to sample with AD converter. I use stanford amplifier SR-570.(~$2K)

- AD-DA converter: NI6110, National Instruments ($3K). This is both for scan control and data acquisition.

- Photomultiplier: Hamamtsu R3896 Select for QY>20%, Dark Current < 15nA. 4 PMT for 1 microscope. (Gree/red channels for epi/trans fluorescence)

FRAP / FDAP in a dendrite

FRAP and FDAP
To track diffusion of proteins in a cell, one can tag proteins in a particular region in the cell, and see how they spread from the region. One way to "tag" proteins in a area of interest is to label protein with fluorophores and photo-bleach the fluorophores in the area using a high power laser. From the time course of the fluorescence recovery after the photo-bleaching (FRAP), one can measure the diffusion constant of the protein.One problem of this method is photo-bleaching of fluorophore produces free-radicals in the cell, causing some damages to the cell.

Another way is to tag proteins is photo-activation of photo-activatable fluorophores such as photo-activatable GFP (Fluorescence decay after photoactivation = FDAP). This method should cause less damages to the cell, and give higher signal-to-noise ratio.

FRAP / FDAP in a thin dendrite
When FDAP/FRAP methods are applied in a thin dendrite and whole section of the dendrite is photo-activated (or bleached), the fluorescence change (Δ F) can be expressed as a convolution of 1D-diffusion

Δ F(t, x) ~ t^-1/2 exp(-x²/4dt)

with a square function with the length of the photo-bleaching area (-b/2 < x < b/2). This results the following equation:

Δ F(t, x) ~ 1/2 [erf{(x+b/2)/(4dt)^1/2} - erf{(x-b/2)/(4dt)^1/2}]

Thus, the time course of the fluorescence at the center of the bleaching region is given by

Δ F(t)/ F(0) = erf(b/(16dt)^1/2)

Note that the shape of the curve is very differen from exponential curve.

FRAP/FDAP in spines

The single compartment model can be used when FRAP/FDAP methods are applied in spines. Because the diffusion inside a spine reaches equilibrium quickly and diffusion between spine and dendrite is much slower, the fluorescence change can be expressed as

dF / dt = - 1/τ F
or
Δ F ~ exp(-t/τ)

where τ is the diffusion coupling time constant between a spine and the parent dendrite. τ should be proportional to volume of spine (V), length of the neck and inverse the neck diameter.

Neurons to make decision

Optical Imaging of Neuronal Populations During Decision-Making (from Kristan's lab)

Understanding how a neuronal network makes a behavioral decision is one of a major goal in neuroscience. In this paper, Briggman et al. imaged the activity of a population of leech neurons using voltage-sensitive dye when the leech makes decision of swimming vs. crawling. They excited the (reduced) animal repeatedly with identical stimuli that stochastically produce crawling and swimming with roughly equal probabilities. Principal component (PC) analysis showed that the activity patterns of the cell population diverge toward two different region in the PC space earlier than the actual response. They identified neurons contributing highly to the discriminant in a linear-discriminant analysis. Interestingly, modulating voltage of one of the identified cells (cell 208) by an electrode biased the decision-making.

Calcium imaging to analyze circuit

Functional imaging with cellular resolution reveals precise micro-architecture in visual cortex (2005), Ohki K. et al.Nature 433, 597 - 603 (Reid lab.)

Congratulations, Ohki-san!!!!! The column structure in the cortex has fascinated many brain researchers: why are cells that response to similar stimulations clustered to form a column ? Using a combination of bulk loading of AM-calcium sensor and a two-photon microscope, Ohki et al. meausred the orientation selectivity of the visual cortex at a single-cell resolution. The sharpness of the boundary between two orientation columns is almost shocking. Supposedly the input orientation map is much broader as dendritic arbor spreads over hundreds of microns. It would be interesting how a single cell process inputs to "choose" one (or sometimes two) selectivity. Another interesting aspect of the orientation selectivity is that mice have poorly organized structure in contrast to cats. To perform complicated information processing, it may be required for animals to have higher organization of a column.

Spine structural change - Nothing to do with physiology ?

Today's journal club was about Shrinkage of dendritic spines associated with LTD, Neuron. 2004 Dec 2;44(5):749-57. from Poo's lab.

Blocking of cofilin phosphorylation blocks spine shrinkage without affecting LTD, and blocking PP1 blocks LTD without blocking spine shrinkage. Both are blocked by calcineurin blocker. These suggest the LTD inducing stimulation activates the calcineurin pathway, and the pathway branchs out to produce two different phenomena, LTD and spine shrinkage.

Clearly spine structural change is NOT required by the change in synaptic strength. This is probably the same for LTP and spine enlargement: I heard from several guys that the washout by patch-clamping often blocks selectively the spine enlargement but not LTP.

Then why the spine structure changes ? One possibility is the structural change is just a side effect: because the actin polymerization is regulated by a balance of kinase/phosphatase activity, LTP/LTD inducing stimulation, which changes this balance, can change the sturcture of spines. Or, maybe this is important for late-phase LTP: larger spines can contain more resource to maintain LTP.