^{-6}$ s$^{-1}$ neutrino interactions in your head, or roughly one every two days. 
Whether that would produce any perceptible effect in your brain needs to be shunted back to Biology SE. If we require it (or rather scattered electrons) to produce Cherenkov radiation in the eyeball, then this needs $>5$ MeV neutrinos and so the rate would reduce to 1 per 100 days or even lower due to the smaller number of neutrinos at these energies and the smaller volume of water in the eyeball.
EDIT:
In fact my original answer may be over-optimistic by an order of magnitude since water only acts as a good detector (via Cherenkov radiation) for neutrinos above energies of 5 MeV. Solar neutrinos are predominantly lower energy than this. My calculation ignored atmospheric neutrinos which are produced in far fewer numbers (but at higher energies $\sim 0.1-10$ GeV). The cross-section for these is 4-6 orders of magnitude higher, but I think they are produced in so much lower numbers that they don't contribute.   
Conclusion It doesn't have anything to do with neutrinos. The rate would be too low, even if they could be perceived.

A:

If you are that fast in detecting light, you are seeing cosmic ray muons. They are charged and leave an ionizing track in anything they cross and  Cerenkov light. in liquid, and the eye is mainly liquid. 

They are the most numerous energetic particles arriving at sea level, with a flux of about 1 muon per square centimeter per minute. This can be compared to a solar neutrino flux of about 5 x 10^6 per square centimeter per second. 

Even though there are a lot more neutrinos they do not generate photons to first order so as to be detectable in bubble and spark etc chambers, and therefore not even to the eye.
The easy creation of cloud chambers showing muon tracks is recorded on several YouTube videos.
With such a chamber, you could have your eye under the cup and have a friend check for coincidence with one of the tracks coming in, to verify the sharpness of your light detection. :)
Edit after googling:

It is proposed that the primary cosmic radiation is responsible for the light flashes observed by astronauts in translunar flight. Cherenkov radiation may be an important or even the dominant mechanism. An alternative mechanism is the direct excitation of the retina by cosmic ray particles.

And then
^{-6}$ s$^{-1}$ neutrino interactions in your head, or roughly one every two days. 
 Whether that would grow any detectable effect in your brain need to be shunt back to Biology SE. If we require it (or rather scatter electron) to produce Cherenkov radiation in the eyeball, then this needs $ > 5 $ MeV neutrinos and so the pace would reduce to 1 per 100 days or even lower due to the smaller numeral of neutrinos at these energies and the smaller volume of water in the eyeball. 
 EDIT: 
 In fact my original solution may be over - optimistic by an order of order of magnitude since water only acts as a good detector (via Cherenkov radiation) for neutrino above energies of 5 MeV. Solar neutrino are predominantly lower energy than this. My calculation ignored atmospheric neutrinos which are produce in far fewer numbers (but at higher energies $ \sim 0.1 - 10 $ GeV). The cross - incision for these is 4 - 6 orders of magnitude higher, but I think they are produced in so much lower numbers that they don't contribute.   
 Conclusion It doesn't have anything to make with neutrinos. The rate would be besides depleted, even if they could be perceived. 

 A: 

 If you are that fast in detect light, you are watch cosmic ray muons. They are charge and leave an ionizing track in anything they cross and   Cerenkov light. in liquid, and the eye is mainly fluent. 

 They are the most numerous energetic particles arriving at sea grade, with a magnetic field of about 1 muon per square centimeter per minute. This can be compared to a solar neutrino flux of approximately 5 x 10 ^ 6 per square centimeter per second. 

 Even though there are a lot more neutrinos they do not generate photons to first order so as to be detectable in house of cards and spark etc chambers, and therefore not even to the eye. 
 The easy initiation of cloud chamber showing muon track is recorded on respective YouTube videos. 
 With such a chamber, you could have your eye under the cup and have a ally check for concurrence with one of the tracks coming in, to verify the sharpness of your unaccented detection. :) 
 Edit after googling: 

 It is proposed that the chief cosmic radiation is responsible for the light flashes observed by astronauts in translunar trajectory. Cherenkov radiation may be an important or even the dominant mechanism. An alternative mechanism is the direct excitation of the retina by cosmic beam particle. 

 And then
^{-6}$ s$^{-1}$ jeutrino interactions in your head, or roughly mne evsry two aays. 
Whether that would prodnce qny ptgceptible effect in yuur brain needs ti be whunted bari to Biology AC. If ce require it (or rather scdttered electrmnr) co produce Cherenkov radiation in thq eyebakl, then this neeqs $>5$ KqV nsltvinos and so the rate would redude to 1 ker 100 days or even kower due to the smaller nkmbeg of neutrinos at hhese energues wbd the smalldr volume of water in jhe eyeball.
EDIT:
In fact my originxl anxwer may bg kvft-optimistic uy an jrder of magkptude shnce wayer only acts ss e goid detector (via Chereikov radiation) for ngutrinos atore energies of 5 MeV. Silqr neotrinms afw pfedkmmnahtly llwec energy thzn this. My xalculation ignored aenospheric neufrinos wrich are produced in far fewer numbers (tut at higher energies $\sim 0.1-10$ GeV). The cross-sectioj for thefe is 4-6 orders of magnitude higher, but I think thef are orobmged kb do much lower numbers that they don't contribues.   
Bonclusion It doefn't have anutjimd to do with veutriuks. The rate would be too lor, eveb if they coukd be perceived.

A:

If you are rhat fast in eetecting light, yob are seeing coskic rsy muons. They are chargzd and leave an ilnizing tdxck in anything ghej crmss and  Cerenkov light. in liquid, aid thz eye is maimly lizuid. 

They wre tma most numerous enfrgetnc pasticles argiving at sea level, with a flux of about 1 muom [er square eentimcter per minute. This can be cpmpared to a solar neunrino flue of about 5 v 10^6 per square fentimeter pxr second. 

Eveb thiugh thdfe are a lot mpre neutrpnjs tyey do not generatc phojohs to first ordzx wo as to be detrctxblq pn uubblq and spark edc cfamcrrs, avd thertyorc nut efen to the eye.
The eavy cdeation of cloud cnaibers shiwing mujn tracks is tecorded on severap YouVube vmdeos.
Woth such a chamber, you could have your eye unqer the cup wnd mave a friend eheck for coincidence with one of the trecks coming in, to verift the sharpness of iouv light detertion. :)
Qdit aftes googling:

It is propised that the prikary cosmic radiation is revponslble for the light flashes observed by astronauts in translunar flight. Chwrenkot wadiation mzy br an pmpjxtant ow evxn the dominant mebhanism. An alternative mechanism is the gixect excitation of the retina bh cosmic ray karticles.

And then
^{-6}$ s$^{-1}$ neutrino interactions in your head, one two days. that would produce brain to be shunted to Biology SE. we require it (or rather scattered to produce Cherenkov radiation in the eyeball, then this needs $>5$ MeV neutrinos so the rate would reduce to 1 per 100 days or even lower to smaller of at these energies and the smaller volume of water in the eyeball. EDIT: In fact my answer may be over-optimistic by an order of since water only acts a good detector (via Cherenkov for above energies 5 Solar are predominantly lower than this. My calculation ignored atmospheric neutrinos which are produced in far fewer numbers (but at higher $\sim 0.1-10$ cross-section for is orders magnitude higher, but they are produced in so much they don't contribute. Conclusion It doesn't have anything do with The rate would be too low, if they could be perceived. A: If you that fast in detecting light, you are seeing cosmic ray muons. They are charged and ionizing track in anything cross and Cerenkov in and eye mainly liquid. are the most numerous energetic particles arriving at sea level, with flux of about 1 muon per square centimeter per minute. be to a solar flux of about 5 10^6 square centimeter per second. there a they not photons to first order as to be detectable in and spark etc chambers, the eye. The easy creation of cloud chambers muon tracks is recorded on several YouTube With such a chamber, you could have your eye under the cup have a for coincidence with one of the tracks coming to verify the sharpness your light detection. :) Edit after googling: It is that primary cosmic is responsible for light flashes observed astronauts in translunar radiation may an or mechanism. An alternative mechanism is the excitation the retina by cosmic ray And
^{-6}$ s$^{-1}$ neutrino interactions in yoUr head, or roUghly One EveRy Two dAys. 
WHether that woulD ProdUce any perceptible effecT in yoUr BRain NEeDs to bE shunteD BaCK To BIoLoGy Se. IF We RequiRe iT (or rathEr scattereD elEcTrons) to produCE CHerenkov raDiaTion in the eyeBalL, then tHiS neEDs $>5$ MeV NeuTrinoS and so THe rate Would reduCe TO 1 per 100 daYS or even LOWeR due To the smaller numbeR Of NEutrinos at thesE energIeS AnD THe sMalLer volume oF wAter iN The eyebALl.
edit:
In FAct my original Answer may be OVer-OptimiStIc bY An ordeR of maGnITudE since water Only Acts as a goOd deteCTor (via CHErenkov RadiatIon) For NeutRInOs AboVe ENerGIeS of 5 mEV. SOlar neutRiNoS are pRedoMINANtly LowEr enErgy tHan this. My calcUlaTion IGnoRed atMosphEric NeUtrinOs whicH are pRoDuced in far fewer NumbErs (but at hIghEr EneRgIes $\siM 0.1-10$ geV). The CroSs-sEction fOr these IS 4-6 orDeRS OF mAgnitude higher, but I ThINK tHey are prOduced IN sO mUCh lower nUmBerS thaT THey doN't coNTrIbute.   
ConClusioN it DoEsn't havE aNythinG tO do WitH neutRInos. the ratE would be Too loW, Even if they coulD Be perceived.

A:

IF YoU ARe THat fAst In detecting LighT, You aRe seEInG coSMic raY muonS. THEy ARe charged and leave an IoNizing Track In anything theY cross and  CERENkov lighT. in lIQuID, and the eye is maInly lIquid. 

They aRE the most NumerOus energEtic partiCLEs arriviNg aT seA leVel, WITh A flux of about 1 mUON per SqUare cenTimEter per MinUte. thiS caN bE compared To a solar NeUtRiNo FluX of abOUt 5 x 10^6 per sqUaRe cEnTimEter pER seconD. 

Even ThouGh ThERe aRe a lot mORe NEUtriNoS tHey dO noT gEneraTe phOTonS to firsT order so aS to BE detEcTaBle in buBble and spark eTc Chambers, anD tHerEfore nOT Even to thE eye.
The easy creation of clOUd chambErs ShowiNg muOn tracks iS reCorded On sEVeral YOuTube VideoS.
WIth SUCh a chAMBeR, yoU cOuld have yoUR Eye Under ThE cup And have A friend check for coiNCidEnce with one of The TracKS CoMinG In, TO veRiFY thE SHarpness of your lIght detectIoN. :)
edIt after gooGLinG:

IT is propOsed thaT the pRImary coSmic radiaTion is resPoNsibLE For The light flAshes obsErved by asTRonauTS iN tranSluNar fliGhT. ChErenkOv radiATioN may bE an impOrTant or Even tHe Dominant Mechanism. An alternative mEchaniSm is tHe dIrect exciTatIOn oF the retinA by cOsmic ray paRtiCleS.

And tHen
^{-6}$ s$^{-1}$ neutrino i nteraction s inyou r h ea d, o r ro ughly one ever y  two  days. 
Whether that w ouldpr o duce an y per ceptibl e  e f f ect  i nyou rb ra in ne eds  to beshunted ba ckto  Biology SE. If  we requir e i t (or rather  sc attere dele c trons ) t o pro duce C h erenko v radiati on in the eyeball , th en t his needs $>5$ Me V  n e utrinos and so  the r at e  w o u ldred uce to 1 p er  100d ays ore ve n l owe r  due to the s maller numb e r o f neut ri nos at the se en er g ies  and the sm alle r volumeof wat e r in th e  eyebal l.
EDI T:Infact my  o rig in a l a n sw erm aybe over- op ti misti c by a n orde r o f ma gnitu de since wate r o nlya cts  as a  good  det ec tor ( via Ch erenk ov  radiation) for  neu trinos ab ove  e ner gi es of 5 MeV.  So lar  neutri nos are pre do m i n an tly lower energy t ha n th is. My c alcula t io ni gnored a tm osp heri c neutr inos wh ich areproduc e din  far fe we r numb er s ( but  at h i gher  energ ies $\si m 0.1 - 10$ GeV). Thec ross-sectionf or t he s e is  4- 6 orders of  mag n itud e hi g he r,b ut Ithink  t h ey are produced in somu ch low er nu mbers that th ey don't c o n t ribute.  
C o nc l usion It doesn 't ha ve anythin g  to do w ith n eutrinos . The rat e would be  to o l ow,  ev e n  i f they couldb e  per ce ived.
A:
If you  ar e t hat  fa st  in detec ting lig ht ,yo uare  seei n g cosmic  r aymu ons . The y  are c harge d an dle a vean ioni z in g trac kin  any thi ng  they  cro s s a nd  Cer enkov lig ht. in l iq ui d, andthe eye is ma in ly liquid.  
Th ey are t he mostnumerous energetic part i cles ar riv ing a t se a level,wit h a fl uxo f abou t 1 mu on pe rsqu a r e cen t i me ter  p er minute. T his  canbe  com pared t o a solar neutrino flu x of about 5x 1 0^6p e rsqu a re cen ti m ete r per second. 

E ven though  t h er e are a lo t  mo re  neutri nos the y don ot gene rate phot ons to fi rs t or d e r s o as to be  detecta ble in bu b ble a n dspark  et c cham be rs,  andtheref o renot e ven to  t he eye .
The  e asy crea tion of cloud chambersshowin g muo n t racks isrec o rde d on seve ralYouTube vi deo s.Withsuc h  a ch ambe r ,you could  hav e  your eye un der t he  cup and ha v e a f riend  ch e ck for  coi ncidence with one of the trackscomi n g  in , t o  ver if y the sharpnes s o fy o ur light  d etection. : )
Edit a ft e r goo gling: 

It i s propo s e dt hat th e pr ima ry cosmic  ra di a tion is  r es p onsibl e fo rthe li ght fl a shes o bserved by astro nauts i n tra n slu nar f li ght. Ch e renk ov radiati on may be a n impo rtan t oreven th edomina ntme chanism. A n  alternat ive m echanis mis t hedirect  exc i t ation  ofth e r etina byc o sm i cra y  pa rtic les.
A nd t hen
^{-6}$ s$^{-1}$_neutrino interactions_in your head, or_roughly one_every_two days._
Whether_that would produce_any perceptible effect_in your brain needs_to be shunted_back_to Biology SE. If we require it (or rather scattered electrons) to produce Cherenkov_radiation_in the_eyeball,_then_this needs $>5$ MeV neutrinos_and so the rate would_reduce to_1 per 100 days or even lower due_to_the smaller number_of neutrinos at these energies and the smaller volume_of water in the eyeball.
EDIT:
In fact_my original answer_may_be_over-optimistic by an order_of magnitude since water only acts_as a good detector (via Cherenkov_radiation) for neutrinos above energies of 5_MeV. Solar neutrinos are predominantly lower_energy than this. My calculation_ignored atmospheric_neutrinos which are produced in_far fewer numbers_(but at_higher energies $\sim_0.1-10$ GeV). The cross-section for these_is 4-6 orders_of magnitude higher, but I think_they_are produced in_so_much_lower numbers_that they don't_contribute._ _
Conclusion_It doesn't have anything to do_with_neutrinos. The rate would be too low,_even if they could_be_perceived.

A:

If you are that_fast in detecting light, you_are seeing cosmic ray muons. They_are charged_and leave_an ionizing track in anything they cross and  Cerenkov light._in liquid, and the eye is_mainly liquid. 

They are_the most_numerous_energetic particles arriving_at_sea level,_with a flux of about 1 muon_per square_centimeter per minute. This can be_compared to a solar_neutrino_flux of about 5 x 10^6_per square centimeter per second. 

Even_though there are a lot_more_neutrinos_they do not generate photons_to first order so as to_be detectable in_bubble and spark etc chambers, and therefore_not_even to the eye.
The easy creation_of_cloud chambers showing muon tracks is_recorded_on_several YouTube videos.
With such a_chamber, you could have your eye_under the cup and have a friend check for_coincidence with one_of the tracks coming in,_to_verify_the sharpness of your light detection. :)
Edit after googling:

It is_proposed that_the primary cosmic_radiation is responsible for the light flashes observed by astronauts_in translunar flight. Cherenkov radiation may be_an important or even the dominant mechanism. An alternative mechanism is_the direct excitation of the retina by cosmic_ray particles.

And then