Thank you Mr. Speaker. I am deeply honoured and proud that the Royal Canadian Legion Branch 258, a vital institution within Highland Creek Community. The Highland Creek Branch was founded back in 1934 and has advocated for veterans and their families ever since. They have always led a successful poppy campaign and this year was no different. They have actively supported local caucuses by sponsoring local cadets as well as hockey and baseball teams and they have contributed to many charities that is because they believe strongly in Canada and its values. More than that, the Legion has always served as a place for longtime friends and neighbours to gather, have a drink and a good laugh. Four years ago when it was newly elected in PP, they welcomed me with open arms and for that I will always be grateful and ever since Mr. Speaker, we have made it a tradition that I come every cannery with a freshly baked cake. In many ways, it is one of my favourite events of the year and this year the old Legion Hall was closed and the Branch 258 has moved to its brand new location at 305 Moorish Road. I know this is a bittersweet moment for many of us but I am confident that they will share many good times in the years ahead. Thank you Mr. Speaker. Member statements. The Member for Windsor West. Yesterday marked the start of Living Wage Week, an important reminder to all of us to recognise the need for a livable wage in Ontario. The Ontario Living Wage Network released their yearly living wage rates for the province for 2022. The stark reality is that there isn't a single one of the 51 communities analysed where the livable wage has been identified under $18 an hour, far above the current minimum wage. Many people in Ontario are finding themselves fully employed yet unable to afford the cost of living as inflation rates continue to soar. The cost of groceries, housing, utility bills and transportation have all increased while wages have remained stagnant. We know that those making the lowest wages in this province are the most impacted when the cost of living increases. In my community of Windsor, a livable wage has been identified as $18.15. That's the lowest hourly wage someone needs to make to be able to put a roof over their head and food on the table. Yet minimum wage is currently set at $15.50, a $2.65 an hour difference equaling $5,500 per year that people are coming up short. Yet the Conservatives were just bragging about their minimum wage. That doesn't account for the employees that are forced into precarious part-time positions where hours are not guaranteed. Many Ontarians are working two or three jobs just to make ends meet. Ontarians deserve to be able to afford the basic cost of living. It's time for big corporations who make record profits this year to step up and pay a fair wage. We must support Ontarians and raise the minimum wage. The Conservative government's empty slogan working for workers doesn't cut it. If they genuinely supported workers, they'd ensure they get paid a living wage. Poverty is this government's policy choice. Member Statements. The Member for Sarnia Lampton. Thank you. It's an honour to rise in the Legislature today to share the news of another important investment in Sarnia Lampton by the Government of Ontario. Mr. Speaker, on October 6th, I was honoured to join the members of the St. Clair Catholic District School Board family along with the Indigenous partners from the Kettle and Stony Point and Walpole Island First Nation at the site of the new Gregory A. Hogan Catholic School in Sarnia. This new state-of-the-art facility which will include spaces for 659 students and a five-room childcare centre is being made possible by an investment of over $24 million by the Ontario Government. This new school will provide more student spaces in the new quality learning environment for the growing student population in Sarnia Lampton as well as affordable childcare spaces for those local parents. In total, the Government of Ontario is delivering more than $26.6 billion in education funding for the 2022-2023 school year, including an increase of over $600 million this past September, which is the highest investment in public education in Ontario's history. Mr. Speaker, the funding for the new Gregory A. Hogan Catholic Elementary School is great news for our community. This investment will ensure families and students have access to a quality learning environment in the years ahead and that every young person has the opportunity to reach their full potential. Thank you. Member Statements. The Member for Scarborough South West. Thank you very much, Speaker. Speaker, Canada is facing an enormous labour shortage. In fact, recent data from Statistics Canada shows that the unemployment-to-job vacancy ratio in Canada is at a historically low level. Here in Ontario, for example, our healthcare sector is a prime example where we are seeing extreme labour shortages, which has added to the healthcare crisis, especially as we try to tackle the challenges of overcapacity in our emergency rooms and to the insufferable long wait times for treatment. According to the Ontario Chamber of Commerce, more than 60% of their businesses and workplaces are facing difficulty with hiring, whether it's healthcare retail constructions just to name a few. Our province simply does not have the human resources to fill this gap. The Minister of Labour himself said that there are over 370,000 jobs across the province, but not enough people to meet the demand. This demand speaker is continuing to rise, with continuing increase in retirement and a slowdown of immigration, especially during COVID-19 owing to the massive immigration backlog on the federal level. And we are seeing thousands of jobs go unfilled. Immigrants contribute greatly to every sector of our province, be it academic, scientific, cultural, manufacturing, food and agriculture, healthcare and many more. I am calling on the Premier and the Minister of Labour Immigration Training and Skills Development to work collaboratively with the federal government to improve Ontario's immigration process and address the massive immigration backlogs so that those who want to contribute, Speaker, to this province are able to do so and have the opportunity to build their life in this great province. Thank you very much, Speaker. Thank you, Member Statements, the member for Brantford Brandt. Thank you, Speaker. As many of you are aware, my riding of Brantford Brandt is home to a vibrant, motivated and active cadet movement. We are home to Admiral Nellis, the Royal Canadian Sea Cadet Corps, the 104 Starfighter Air Cadets, the 56 Field Regiment Army Cadets and the Admiral Landymore Navy League Cadet Corps. After a two-year hiatus, this remembrance day, cadets representing all four corps held a nighttime vigil at the Brant County War Memorial. I stopped by the DeLuzzi Street Senate to have to spend some time with the cadets, officers, volunteers and family members as the youth stood in silence, paying tribute to Canadian and Allied soldiers that fought, were wounded and paid the ultimate price for our freedom. I spoke to several officers who were supervising the cadets and Lieutenant James Massacar, commanding officer at the Royal Canadian Sea Cadet, said, the remembrance vigil is a living act of remembrance that the young people of Brantford, Brant and Six Nations take part in. It has endured since 2011 not only because of the commitment of the community, local leaders, officers, but most importantly the sea, navy league, army and air cadets who hold the torch of remembrance high each year as they mount their vigil. Speaker, witnessing firsthand the solemn respect and sheer professionalism these youth displayed for the fallen soldiers of our country far exceeded their young years. I was moved and impressed. Brantford Brant has some of the best volunteers in Ontario and I urge all to support your legions in Cadet Corps and encourage them to keep up the great work they do week after week. Thank you, Speaker. Member statements? The Member for London West. Thank you very much, Speaker. London West parents are worried. One mother wrote to me, parents are looking for guidance from our leadership right now and we are getting nothing. As a parent to a three-year-old son I am terrified when I see the news about pediatric ICU beds. Another said, the state of our healthcare system particularly pediatrics is horrifying as the mom of a toddler who has been sick with COVID, hand foot and mouth disease and pink eye in the past six weeks I am terrified. While we have been lucky to not have to go to the ER yet I am fearful of what we would experience when we arrive. What she will experience at London's Children's Hospital is a stressful hours long wait in a crowded emergency room that was built to handle about 100 visits per day but is being overwhelmed by 200 or more sick children double the usual volume. Parents of teens admitted to ICU now face the prospect of admission to an adult ICU bed which has ICU nurses concerned about taking on teen ICU patients without specialized pediatric training. Children's Hospital emergency room director Dr Rod Lim warns that it may get worse before it gets better. I think November and December are going to be tough he said. Speaker London parents are asking me what is this government doing about this crisis from the budget that was tabled yesterday my answer is clearly not enough. Thank you member statements the member for Mississauga Lakeshore. Thank you Speaker on Friday last week the President of the Treasury Board joined me in Mississauga Lakeshore for a tour of the construction site at the Mississauga Hospital where work is well underway on a new eight-story parking structure with spaces for almost 1500 vehicles. This is an important first step towards the complete reconstruction of the entire hospital the new 24-story facility will almost triple the size with almost three three million square feet of space and almost a thousand hospital beds over 80 percent which will be private rooms it will add over 20 new state-of-the-art operation rooms this will be the largest and most advanced hospital in the history of Canada. Speaker the fall economic statement yesterday reports that the RFP has closed and it is confirmed our government is committed to this historic project so I want to thank the Minister of Finance for his statement yesterday and this Minister of Finance cares about Mississauga Lakeshore and again I want to thank him and the President of the Treasury Board for their support for this important project. We've been looking forward to a new modern hospital in Mississauga for many years and it's very exciting to see the shovels in the ground. Thank you. Member Stevens, the member for Don Valley East. Thank you Mr. Speaker. Mr. Speaker I stand before you with the grim realization our children are being let down by this government. It has been said that the true measure of any society can be found in how it treats its most vulnerable members. All of us bore witness to this government's failure of our elders in long-term care homes during the pandemic. Most recently this government took aim at our lowest paid education workers by trying to trample on their charter rights and now is our children who our government is failing and they don't have a voice to advocate for themselves. Regrappling with overcapacity pediatric ICUs, ERs in critical care that is overwhelmed, kids in adult ICUs and surgeries being delayed. This summer when this government was nowhere to be found I called for the government to take measures to prepare for the upcoming respiratory season and yet the fall vaccination campaign was non-existent and the plan to stay open has not eased the emergency care crisis in the in the slightest. Now I would be remiss if I did not address the current state of the Ontario autism program. Under this government the OAP's waitlist for core services has more than doubled skyrocketing from 23,000 in 2018 to 57,000 in 2022. That is enough children to fill the Roger Centre and then some. This government needs to be transparent with families. How many children are registered? How many are waitlisted? And how many are receiving core services? I eagerly wait for those answers. Thank you. Members statements? The member for Thunder Bay Atacoken. Speaker I recently had the opportunity to attend a fundraising dinner for the medical equipment modernization opportunity or memo for short in Thunder Bay. Since 2004 memo has shipped 100 containers to Cuba, El Salvador, Liberia, Nicaragua and Zambabwe totaling millions of dollars in equipment and saving countless lives along the way. Every container holds 10 tons of medical equipment and supplies that are redundant according to the Canadian health standards but remain useful and continue to save lives in developing countries. The organization is completely run by volunteers so that every dollar donated goes entirely to support the work. I want to recognize Dr. and Mrs. Jerome Harvey whose dream began 18 years ago while looking outside of their kitchen window at the soon to be closed 250-bed Port Arthur General Hospital. He wondered what was going to happen to all the medical equipment left behind in this hospital as well as the other old hospital to be closed, the 375-bed McKellar Hospital. He discovered that because of Thunder Bay's isolation the cost of removing and shipping made the contents of the two old hospitals valueless. Reluctantly the hospital administration admitted that most of the equipment would be sold to a scrap dealer for $180 a ton. This led to a meeting with the Cuban doctor and Cuban ministry of health officials and assured him that anything memo could send would be helpful as Cuba had an adequate supply of well-trained doctors but was woefully inadequate in medical equipment and supplies and thus memo was formed under the auspices of the Evangelical Free Church of Canada. I commend Dr. Harvey for his initiative in leadership of this venture that is making a difference in the lives of residents in developing countries. Thank you. Thank you. Member Statements to Member for Don Valley North. Thank you Mr. Speaker. I'm hearing from many constituents in my riding up Don Valley North who are extremely concerned about the dramatic increase in incidents of vehicle theft and carjacking in our area. Speaker, according to an article in The Toronto Star, 70 vehicles were stolen in North York during the week of October 13th to 19th. The previous week the number of stolen cars in the area was 54. Speaker, there is one vehicle stolen every six minutes in Canada and the proceeds from stolen vehicles are often used to fund global organized crimes and terrorism. Speaker, this issue is being addressed by the police in our community and through public safety initiatives in our province but without proper legislation and procedures to hold accountable those who alter vehicle identification and to prevent stolen vehicles from existing Canada we are far from solving this serious public safety issue. Speaker, I hereby call out to the federal government to tighten border controls and stop stolen vehicles from being exported overseas. Thank you Mr. Speaker. Thank you. That concludes our Member Statements for this morning.


I will put presentation mode, sorry. OK. Thank you. Thank you. Thanks, Matteo. And good afternoon. Today, I hope I will cover a different aspect of time encoded in neural activity, but we will move from ramping to overall activity of the network and network dynamics. Anyway, so that's why I just wanted to start it from this slide, which is basically the summary of what we saw yesterday. So we saw that time modulates the activity of neurons. Here are 58 neurons in LIP. So basically, time is expressed in this slow or fast ramping of the neurons. And we saw this in animals, but we also appreciate that even in humans at a different, of course, special scale. And we observed not only in parietal cortex where we saw this neural activity, but also in other brain regions. So we saw that time can modulate its activity in a wide network of brain regions, going from occipital cortex to frontal cortex. So in this somehow, our data that experts in the field took as a demonstration that the mechanism is to tell time is in this sort of ramping activity. Other people, this is something that it's put forward by Warren Mack in this review paper, which is published in Nature Review Neuroscience, where he basically talked about striatal bit frequency model. And according to him, basically, the time is processed and is by a specialized circuit that basically sees the involvement of many cortical regions that work as oscillators. So you see here, this is a brain of a human being. So in here is the cortex. So this is what you see represented with these little circles are basically different. He's pinpointing the different cortical regions. All these regions, so this is parietal cortex. This is primary motor cortex, premotor. This is called prefrontal area. And each of these bits of the cortex are supposed to have a different functional property. So to be specialized for something, right? Like the parietal cortex we saw yesterday is an area important to process visual information that then serves for actions. Primary motor cortex is the area that it's responsible for really the actual movements that we make. Premotor areas are active before we actually make a movement. And whereas prefrontal cortex is a part of the brain that is supposed to be engaged in executive function. So they work as a controller that everything goes according to the plan. So they respond, for example, if there is a mismatch between what you are doing, what you're planning to do, and what, for example, the feedback that you get from reality. So this is also good for planning in not just in motor sense, but more in an abstract way. So when you, I don't know, if you play a game and you want to use a certain strategy and you foresee, for example, in chess, what are the movements that you want to make. These are the areas that more or less should be involved in this abstract level of reasoning. And basically what he believes is that in these areas you have oscillatory activity, right? This is also something that has been observed. So that this oscillatory activity that you see represented here in this wave that have a certain face. Of course, the peak-to-peak is the frequency of these waves and the brain can oscillate. So oscillations is due to the synchronous activity of many neurons. This is what produces oscillatory activity that you can record from the outside, even with an electron, with electron cephalography. And so basically he assumes that different brain areas can oscillate a different frequency and what is really the process of telling time is basically guided by this area here that is sub-cortical. It's underneath the cortex. And basically when a certain interval starts, so you see here the representation of the beginning, oh, sorry, the beginning of a temporal interval. So imagine that you have to tell the time that is between two beeps, like beep, beep. So the beeps are equal. What changes is the empty interval between the beeps. This is what makes an interval. And this is representation of what's going on in the brain when you deliver these two sounds and there is an interval between them. So according to this model, this idea, you basically have, whenever you have the first sound, you have activity in this basal ganglia, so in this sub-cortical region, in particular in sub-nuclea is a portion of this called substantial nigra. It's just some nuclei in basal ganglia. And this starts firing at the onset and then this, it's like triggers the clock. Then you have multiple oscillatory activity across the cortex. And then you have some, at the end of the interval, you have, again, another burst of firing in the activity of the basal ganglia. So they just, basal ganglia is supposed to work as a start gun and it's also, so whenever it detects a certain level of synchrony in these neurons that corresponds to a previously rewarded situation, then it fires again because of course, the model assumes that you learned through experience to identify a certain interval, okay? So it's a detection of a certain oscillatory pathway in the cortical regions. And here you see more or less something here where you see that there is trial onset and there is a lot of firing in the neurons and then at the time of the reward, there is this striatal activity. Slowly, Dominica. So in the figure on the left, it looks like the function of this start gun is to essentially reset these oscillators. Exactly, exactly. And to synchronize them, right? Exactly. It synchronize them and it, and it fires again when it detects a pattern that has been seen before. So that has been detected before as a consequence of a reward. So it synchronizes and then it's just to stop when a certain pattern, I mean, I'm sorry to be so imprecise, but in the end, this guy is imprecise. So this is what they say in the paper. So it's very, it lacks of details. So it's unclear why all these regions to start to synchronize in the first place, why you should see a lot of synchrony activity across regions that have a very different function and how do you get, how do you learn this through experience? This is also unknown. What is the reward? Okay, so this is, it's just a very descriptive, let's say way of presenting a possible mechanism for temporal processing. But what I, for me, it's important is for today's purpose is to highlight the basal ganglia. So the function of this area that is underneath the cortex, that is an area that it receives information, here you see it in a very schematic way, but communicates with most of the cortex. So it's an area that it's widely interconnected with many cortical regions. And it's an area that it's known for motor coordination. So we know that, for example, Parkinson's patients that have lesions in basal ganglia that have motor problem as well as cognitive problems. But one of the first signs is just the lack of, for example, they become very slow in moving, okay? They are difficulty in synchronizing movement. And also what it's very important, so it's an area that becomes active when you expect a reward. So there is an expectation of something that should happen. And this area, the activity in this area is modulated. For example, let's say you train, because I'm talking about something that has been studied in monkeys, for example, you train a monkey to associate the presentation of a sound with the delivery of some reward like fruit juice to the monkey, right? And there is a certain time between the sound and the reward delivery, okay? So at the beginning, so before training this neurons in basal ganglia starts firing immediately after the reward, the monkey receives the reward. So, okay, so the response to reward to, but if you then, if the monkey learn to associate the sound to the reward, then this neurons starts firing at the, when the sounds is presented. So before actually the reward occurs. So they kind of predict the reward occurrence. And if the reward is not given, then activity in these areas it's also modulated by the lack of something that is expected, okay? So if there is some mismatch between the expectation and what they get, then also this activity in basal ganglia is suppressed. So it's an area that it's very important for predictions. And this can be very useful also. We said yesterday for time. Anyway, so now I want to show you a study. Again, it's an electrophysiological study. Here they use mouse, so mice, so two not rats. And it's an electrophysiological study where they train basically mice to really do a discrimination, a purely perceptual task. So far, most of the study that I presented to you that in animals, they often use reproduction tasks. So task where the animal had to sort of reproduce with the movement or all being a movement to express the judgment of time. Here it's a really more perceptual task. And I will explain what I mean with that. So basically the trial starts when the mouse insert the nose into this nose poke, okay? So the trial starts. And then they receive basically a first sound of a certain duration. And then there is an interval where nothing happens. And then they receive a second sound that has a different duration. And then the rat, sorry, the mouse, has to decide which one of the two was longer. And they have to basically put the nose into the nose poke either, I don't know, so they are trained if they do like leftwards. So if for long, so if they judge long and rightwards for short, okay? So they learn which nose poke to go to express their judgment by receiving a certain reward. So here, for example, in this picture, they have been rewarded for judgment long if they go leftwards and for short, if they go rightward, okay? And the reason why I'm talking about this study now is to show that there is a different way of looking at neural activity and the way neural activity can express time. And how also neural activity can be linked to perception, okay? Here we have, this experiments the rat with the mouse with its response express its perceptual judgment, right? So and what they do, they target as an area where to implant electrodes, the basal ganglia, in particular the dorsal striatum is called. It's this area here, this is a mouse brain. So of course, the shape is different from human brains, but here you still see the cerebellum as in humans. This is front and this is back, okay? And this is an area that is below, this is the cortex, the most superficial part of the newest phylogeny, ontogenetically and phylogenetically as well to develop. And so it's an area that it's underneath the cortex, it is this one.  So if they ask you to perform a task. So here again, you do in the Y axis, the percentage of long choice. And this is the on the X axis, you have the actual intervals. So basically, sorry, I didn't say that they have to discriminate between two stimuli. And one of the stimuli last 1.5 seconds and the other can be either shorter or longer than 1.5, okay? So and here you see how the psychometric curves look like, okay? They, I think they train five mice in doing this task. And I think here you see the psychomal, maybe more because here I see more than five lines. And I think in black, it's the average of behavior of this train mice. So you see if you have a very good behavior. So subject, so the mice said very few times then the stimulus, the second stimulus was longer when it actually was shorter where they say when the second stimulus was longer they have more possibility to say longer. So they basically perform the task well. Now, what is also interesting about this study is the fact that they use multiple methodologies to address the question which is basically if so they want to test the role of this substantia nigra. So this striatum in a temporal perception. And the first method is to produce, okay, sorry. First I want to show you the behavior of these neurons, okay? So this is to tell you that the neurons that they, here you have in red the plots of the firing rate of the population when you have incorrect choices when basically the mice are wrong in choosing the nose book because in judging time. And the green are when the response is correct. And the first dashed line corresponds is the first stimulus, the first tone. And the first is the second tone, okay? So these are really what happens during the trial. So during the trial when the mouse experience different events. So you see that those neurons fire for the first tone at the offset of the first tone at the offset of the second tone. And then of course there are some of them are insensitive to reward than others are. So you see this is what I said before, no? That some of those neurons are known to be active at the reward delivery. Now what they do and here it's again same representation of the population. So this is the average and these are responses of the neurons at the two tones, first tone, second tone. And the color code codes the different durations they are perceiving. So they are not perceiving, they are exposed to, okay? For different physical duration from I think the shortest to the longest. So you see that they both react to the two stimuli and more or less the amplitude of the signal doesn't change according to the different duration that they experience. Now what they do and what I was telling you is that the first technique they use, so apart from recording the electrical activity from those neurons, they also, in some of them they inject basically clots up in. So they inject this chemical that it's in some of them. So in a group they inject this clots up in that basically creates a lesion inhibits the neurons in these regions. And in some of them they inject the selling solution. So because they want to compare the performance in the temporal task in those two type of conditions. And this is what you see depicted in the plot. So you see again the behavior against the different interval and you see in black the mice that get selling solution. So you still see a good behavior. But this is the basically the slope for the group that receives this clots up in. So you see that, so in this case those researchers show that this portion of the brain is somehow necessary for the mouse to perform the perceptual, the temporal discrimination task. Okay, now another interesting thing and this is something that I hope you appreciate it now. We go back and forth a little bit. So what I'm plotting here is a very, it's a cartoon that illustrates the different patterns that you should see in behavior. If the changing behavior that you see is due to time or is due to an action bias. Okay, so if you are biasing somehow the clock or you're biasing the decision level. So if what you are producing with your manipulation in that case was injection of a substance that suppress activity in the striatal neurons. If those neurons would create a bias in the decision level, you should see basically this is the pattern in the psychometric curve that you should see. So it means that the mouse starts responding always shorter. No matter the stimulus, it gives more shorter response or more longer response. So this is basically a vertical shift in the psychometric curve because if I'm biased, if I say always short, this is how it look like. If I say always long, that's the other bias. If this is the blue curve, okay? So this is how an action bias should look like because you more often are likely to respond short versus long. If it is a time estimation bias, so the lapses. So what I'm saying is the lapses of this curve are changing, okay? It's not just the curve is shifted towards one direction or the other, but you should see also a change in the lapses of these curves. Whereas if the bias is in the time estimation is not at the decision level, but it's more at the encoding of duration level, then you shouldn't see this, any change in the lapses, but where basically the response is pretty obvious, right? So I don't know if you are able, if you see this, because here if I say, so here I see something that it's definitely physically, clearly very short and I judge it as if it was long, okay? This shouldn't happen if you're, you perform, if you encode the duration well, okay? So because these are pretty, the lapses are the most easy condition to discriminate. So what you see, so what you should observe instead is a basically a shift, sorry, in the only a shift and not a change in the lapses. So if the observed change in performance is more likely due to a time estimation, yeah, it's something that happens at the time estimation level, because you are biasing the behavior in the most difficult condition, okay? When basically the difference between the two stimuli is very subtle and it's here that you can make an impact and you can basically bias the perception. Now what they do these researchers is also to, they don't look at the ramping like the previous researcher, right? They don't search for that, but what they do, they basically try to classify the activity of the neurons based on really the amplitude of this activity overall. This is an example, this is the distribution, this is an example of three different trials in which the activity of the neurons where basically could be very high. So this is the distribution of the response of the neurons, okay? These are, I think, few neurons, it's not the population. And so as you see, there are sometimes that the activity, the amplitude of the signal is pretty high, sometimes it's average, sometimes it's low. Okay? And they basically, and this is the, okay, what they're interested in is to classify somehow the activity as low, medium, and high at the offset basically of the second tone, okay? Because first tone, second tone. And this, they make this classification. And here again, this is again the same plot, but these are only neurons in that response. This is only a trial when the second stimulus lasts 1.7 seconds. But it's the same type of representation. So you see high activity, medium, and low. And what they do basically, they use this sorting of the neural activity to reconstruct, to see the behavior. So basically they, what you see in this plot is again the behavior. So the dots are the actual behavior of the five mice that they test, okay? So this is, and then what they do, they try to see, so to this performance, what type of activity in the striatum corresponds, okay? So they sort the, they see, they check what type of level of activity striatal neurons show for these different types of behavior. And this is what they see basically. They see that low dopamine response corresponds to a leftward shift. So this means that something that lasts, let's say one second is perceived as if it was 1.5 seconds, okay? Because this is what we call, we saw this many times during my lectures, but also at the beginning of my course. This is what you look at is the point of subjective equality. So it's the performance when the performance is a chance when basically the mouse can say either short or long. It has equal probability to say long or short, okay? So this is a measure of bias because, so if the 50% it's here, it means it's in the red line, it means that something that it's physically less than 1.5, so that it's one second is perceived as if it was 1.5. So there is a overestimation of time. And this overestimation of time is associated with low dopamine, sorry, low striatal activity. And if the shift is in the red line, the shift is towards the right, so the blue curve, it means that something that it's physically a bit more than 1.5 is perceived as if it was 1.5. So there is a time under estimation of the mouse. So high activity in the striatum is associated with time under estimation. Why I talk about dopamine here? So lower dopamine and higher dopamine. This requires a little bit of basic knowledge. So dopamine is the neurotransmitter, is the principal neurotransmitter that these neurons use in synapses. So neurotransmitter is a substance that is basically produced by neurons. And it's basically delivered once this action potential is occurs, okay? And the neurons has also the capacity, so it's in the, so the neurotransmitters basically, it's delivered in the synapses and it's basically affects the dendrites. So affects the cells that receives, let's say, the signal. Okay. And the major, there are several types of neurotransmitters that the brain uses. One of these is dopamine. And dopamine is widely presenting these subcortical structures like the striatum, the basal ganglia. Okay, so you see that here there are two key points the fact that you can link the level of activity in a brain structure, which is subcortical and you can link a certain substance loss or dopamine because those structures mainly use dopamine to time perception. You can relate, you can associate dopaminergic activity in the striatum to time perception because according to the higher or lower level of dopamine in the striatum, you can see a bias in the performance of the mouse. Now, in order to be even more convincing in showing this, those researchers use another technique that it's very, it's something new indeed that exists. I think it has been developed in the last 10 years in neuroscience and it's co-optogenetics. I always, I think I told you about this technique.  And they basically, this virus changes basically the neurons by making those neurons sensitive to light. Okay, so they basically, the neurons gets excited or inhibited according to the wavelength that they receive. Okay, so you can basically excite the neurons if the light, if the wavelength is in the range of 500, yes, 556 nanometer or you can inhibit. So the light is in the reddish spectrum whereas you can inhibit the cells if the light, if the wavelength is in the range of 400 nanometers, okay? So this, it's a very, I think you realize because it's a better way to do what I show you in humans you can do with TMS, right here. You can really suppress or excite the neurons from which you're recording. So you can make not just correlational inferences. So by saying, okay, this dopaminergic activity, the higher or lower dopaminergic activity correlates with a certain bias in time perception, but you can really kind of saying that this activity, it's causal important. You can really make also causal inferences here. And what they do very, I think, interestingly is that apply this, so you stimulate with the laser beam really optically. And what they do, they just, they do activation. So they try to excite this striatal neurons and this is what you see in this panel F where you see basically that if you, so this is the performance at baseline gray and black, okay? So without optogenetic, this is the behavior of the mouse without stimulation. And this is when basically the same behavior of the mouse when you apply this optogenetic stimulation and you are basically trying to inhibit, sorry, to activate the neurons, okay? And so when you increase, so this, when you increase dopaminergic activity, you see that the curve shift rightwards. So it means that you create a time under estimation, okay? So because the curve is shifted toward right, so it means that an interval that it's physically more than 1.5 is perceived as if it was 1.5, okay? So you underestimate time. Whereas when if you do inhibition, so if you use a different wavelength, then you create an opposite bias. So you observe the overestimation, although the magnitude of the effects, you see clearly the shift, it's much smaller. But I think overall this is a very comprehensive way of very, way of addressing a question, right? With different approaches. And what I like is the fact that they also have this, so we see at the beginning the inhibition of the activity by a lesion, the clozapine injection, then you have the electrical recording and you see how activity in this, the firing rate can be, the amount of firing rate can be linked to time perception and you just do the optogenetic manipulation to actually prove that this level of dopaminergic activity can shift the performance, can impact the performance, okay? So we move from ramping to considering the overall activity of the striathlon and from parietal cortex to subcortical region. Now I just want to spend the last part of the lecture by showing another experiment, but this is, so I hope it's gonna be very easy for you to follow because it's the same group of people that there is, yeah, there is a question. Yes, can you go back a slide, please? Yeah. So in the psychometric, in the psychometric curves at the bottom of the slide. Yeah, those two. In the blue case, for example. Yeah. The activation, there's a shift towards right. Yeah. So this means that most people will start recognizing at later in time, right? What means is that the mouse, when you see some, so what you measure here is the shift of the curves. It's called, it's a, this shift happens here, right? That corresponds to 50%. So basically your, what you're looking at is what is the stimulus that basically elicit so that leads the mouse to say half of the time longer, half of the time shorter, okay? So basically is the value of the physical duration that you present that basically leads to chance level for the performance is a chance because 50% of the times it's classified as longer, 50% of the time is classified shorter. Now, if you are a perfect clock, this should happen when the two stimuli are 1.5 because what the mouse has discriminated is between two stimuli, okay? One meets 1.5 and the other can be 1.5 or either shorter or longer, okay? So you expect the mouse to have this sort of 50-50 respond when the physical stimulus is 1.5 because this is the truth, right? Because if you present two things that are equal, no? You, you know, and if I force you to tell me which one is longer. No, on these I agree. But then when the shift is longer. Exactly. And when the shift is on the right, it means that this type of behavior, this chance level behavior 50-50, the mouse gives this behavior when a stimulus is not 1.5, but it's bigger. So it's 1.6 or 1.7. It overestimates. No, no, it means that it underestimates because it judges something that it's physically one, something that it's physically 1.7 is judged as if it was 1.5 because it's the point of... I'm using overestimate in the opposite way as you are using underestimates, but yeah. Did you understand what I mean? The logic is this because to me it's, yeah, it's since you're judging something that it's a bit long, that it's physically you judge it as if it was 1.5 because 1.5 is the point of, we call it subjective equality. So it's the point where you perceive two things to be equal. And so your perception, your choice is 50-50. Is it clear? Yes. Now? Yes. It's a bit confusing. Are you totally... There is a question on the wavelength of the laser that you're using green light for this 356 nanometers. Yeah, it's I think a bit closer to yellow-green. This is... Yeah, it's red-green for excitation and blue-yellow for inhibition. I never use optogenetics because I don't know the size of it. We refer to the papers. Yeah, exactly. So this is the wavelength that they use. But basically because, yeah, but even in the paper, you won't find a lot of more details than that anyway. So yeah, you can check. Okay. So what I wanted to now show you is that we are moving in perspective. We considered the whole population activity. So we went from observing a certain type of behavior, the ramping, to just considering the behavior of the whole population. Okay? And I wanted to stress that looking at ramping or a certain type of behavior, it's also very risky. It's not an optimal strategy because if you talk to any electrophysiologist, he would tell you that basically the behavior of neurons, it's very complex. So you don't see... So, okay, you see in a bunch of neurons that they behave in a certain way, but others behave in a totally different way. So that's why I think it's sensible to move from just looking at a certain type of behavior to looking at the more global level, the behavior of the population. I think it makes sense to move into this different perspective. And these guys here are the same, so the same group, Champolimod in Portugal, and they use... Okay, they just change species. They just go into rats, but the task is exactly the same. So the rat this time has to discriminate the two sounds, exactly the same behavior. So long or short, okay? He has to decide, he perceives two sounds, he has to decide which one, so whether it was longer or shorter than a standard interval. This is the psychometric curve again. This is the behavior of the rats. But what it to me now it's interesting is that now they, because also electrophysiology is tricky, depends how you look at things and things can change radically. Now what they, the way they look at these neurons is to see whether there was some selectivity to durations. Okay, so basically you see that the rat perceives gets different durations, no? From 1.5 to 2.4 to 0.6. And there is a full range here in between, okay? So he experienced different durations. And if you look at single neurons, so level, not the population, you see that some neurons like 200 and 2.4 seconds. So this is a, it's called a raster plot. So each row is basically the time of the response of this single neuron. So you see that, so different trials. So in different trials you see that these neurons like a lot, the 200, the 2.4 seconds. And this is the average of this across trials, okay? This is the behavior of this single neuron that spikes, so his peak, he spikes a lot for, sorry, for a duration, I'm sorry, I was just wrong. The stimulus that he was delivered was 2.4. But the guy, so this neuron likes a duration that is 0.6 because his peak here. So at the end of the stimulus, it doesn't peak at all. And here it's the behavior of a totally different neuron, so of a totally different neuron because the name, here is the label of the neuron, 132 and 290. The stimulus is still the same, is a very long sound. But here the behavior, it's opposite to this. So this guy likes very long durations. So here you start seeing duration selectivity in this population. And here what they do, this is basically the authors of the paper. Here they are sorting all the neurons from which they record, it's 433 neurons. They sorted in the firing rate according to the preference of the neurons, according to duration preference of the neurons. So you see, you have this beautiful diagonal where you see that different neurons are selected for different durations. So you see how things are changing here. And here it's the, here is exactly the same plot as I show you, so the same raster plot than I show you before. And this is basically, here you have the distribution of all the cell, the population that they considered. And this is basically a way of seeing how much duration selective neurons they see in the population. So you see there are some, so this is a cell count. This is number of cells that prefers short versus preferred long, okay? Here it's again the same sort of plot that they used so before. So this is a cell that prefers short, this is a cell that prefers long, okay? And so what they do then, and here I think it's also very interesting. Now they look at, they sort the neurons according to the preference to different durations. And here they have a low dimensional plot. So they now consider the whole populations of the neurons they have. And the color code, it's the sorting they have according to, so they have a boundary according to which they classify neurons as having a certain duration preference. And according to this sorting, they try to have a neurometric. The neurometric is this orange line. The psychometric is the green line. So the psychometric is really the behavior of the population. And by sorting basically the neurons according to the duration preference, they build up this neurometric. And as you can see basically this neurometric curve, it's very close to the psychometric. It fits better the highest bit of the psychometric rather than the shorter part, but it's pretty close. And it's definitely close better than doing a classification because here, okay, this is obtained by classifying the response, okay, the green of the, this is by classifying the response by reconstructed by classifying the neural activity. And this is doing a psychometric by classifying the videos. So basically they record the rat when it just goes left or right. So here is a classification of the videos according to the movement that according to the choice basically that the rat makes towards the left or towards the right. And so the videometric you see it's worse, much worse than the other. But the step I want to point out is a step farther to that. So here, so by classifying the activity of the population according to the duration preference is having a snapshot of the population activity, okay? Because you forget how a certain neurons changes is activity over time. So you basically forget about this, the dynamics over time. And so basically they move into this new way of looking at neural activity by now considering this and by sorting now the neurons according to these differences in the dynamics over time. And here again it's lower the dimensional plot. So it's a principal component plot where they plot the different, so they represent let's say this dynamics over time of the different neurons according to the trajectory according to how fast or slow they reach a certain state, okay? And this is basically this is the plot of the average trajectory, the trial density. So you classify the trials according to the trajectory that these neurons cover, okay? So they translate temporal dynamics into a spatial trajectory. So and more trajectory for them means faster speed. And by using this classification that it's made according to the trajectory that the neurons cover over time, they built psychometrics basically, okay? They built and what they seem to show is that so they do the same job. So the dots are behavior and the lines are basically are reconstructed so you sort the red or the blue line according to the fact that the trajectory was fast or slow. And what they see is basically that the more trajectory so the faster speed corresponds to a time overestimation, okay? So which is this one. And slower speed corresponds to time underestimation. So, and... Excuse me. Yeah. I didn't understand properly the previous slide diagram about projection and also this diagram about principle components. How they draw this and how they draw this and how they conclude the neurometric diagram from these... Yeah, this is a yes. Okay, this is a, it's a good question because it's a, you should read the paper actually to understand every single step. It's a, and for me was too complex to tell you but basically, so what they have they have a high dimensional space, no? Because they have all, so imagine that for each neuron you have the behavior of the neuron over time. You have all these neurons, you have each neuron it's classified, so it's classified also according to these different preferences. So you have different things to take into account, right? The preference and how this, so in principle there are, so in principle there are these different populations, right? That they seem, you see here, right? So there is neurons that prefer that different type of the color code means different preference for different durations. So in the first place they have all these and for each, all these preferences they have neurons and they have the neural activity over the time of the experiment. So now basically these spikes, the frequency of the spikes changes over time. And what they do, they do a principal component analysis. So they try to see all the variants of all this, yeah, of all, so the variability of the behavior that they observed in all this data. And that they try to reduce into principal components that are represented here. So you represent in this high space into a trajectory that has just two components, two, you put everything in two dimension basically just for simplicity. And what they, so what they, they try to, to with this principal component, they try to basically say that this, they try to translate this complexity of the, of the dynamics over time into space into how fast these dots go from here to here basically. But you, you have to read the details of the paper. I think it's in the references that you, that I give it to the course. So the axes are referring to space by principal component? Yeah, the principal, yeah, these are the two main major components that they derive from feeding with the multidimensional data that they have. And this is just a trajectory in space basically that translates the, in space, the dynamic that they see over time. This is my understanding of the thing. You, you told something about. So faster means that they change fastly over time. So more trajectory means that they, they, they, so if this covers more trajectory means faster speed that you reach this, you change, yeah, that the network should change more quickly over time. It's faster in changing the state compared to the slower changing population. This is my understanding, yeah? You were saying? One other point, you told that some neurons prefer some time duration more. And you showed us that diagonal diagram. And do I understand correctly that in the brain, each neuron is responsible for perceiving some special time interval? Yeah, I mean, this is what the data seem to show. So that if you have a specific, so this, this, this neuron for example has a, so response mainly to 06, but you see this is a sort of caution like, right? So it's also response to stuff that are closed. So to stimuli that are close to 600, but he has a preferential range to respond. This is correct, yeah. And this is what also we will see even tomorrow. There are other studies that show that you have a certain selectivity to duration that means that if something is physically there for more or less time, you have specific neurons that seem to care about the fires that are active for those intervals. But you have to bear in mind that this selectivity can be plastic and change, okay? So because otherwise it would be a very expensive mechanism, right? So if you have to encode for all possible durations, right? So then there is a, as you see, this tuning has a certain plasticity so it can be, so you can change the tuning. You can shift also this curve, this peak. You can shift the peaks towards the left or towards the right. So this peak, this selectivity is not absolute, but has some degrees of freedom in the sense. But yeah, but these two, yeah. Thanks. This is very fascinating because you wouldn't expect that something that is so abstract somehow, no? Time is so, it's something that you cannot really perceive with the, you cannot sense like a color. It's not concrete, it's not like a color or shape or nodder. It's something less elusive somehow, but it has such a precise representation or elicit, such a precise behavior in a neural level. This is fascinating, yeah. And it's new, huh? So it's not something that we know from a long time. Okay, so Dominica, do we take some more questions? Yeah, if there are, I mean, depends if they have time or if there is another lecture later, I can take some more questions if there are. Okay, so let's see. If there are no other questions, then we take a break and we'll... We'll see you tomorrow, yeah? Yes. The last lecture on duration tuning. Thank you very much. 10...


The Chinese solved cubic equations through a process of successive approximations. They could find as many digits of the root of a polynomial equation as they wanted. But we do want to ask, does this count as solving the equation? We also saw that Islamic mathematicians solved cubic equations by identifying conic sections whose intersection points corresponded to the solution of geometric problems. But again, we might ask, does this count as solving the equation? And the question comes down to what we mean by a solution. For example, consider the equation x squared equals 3. We might give the solution x equals the square root of 3. But what would Chinese and Islamic mathematicians say about this solution? Well, in China, you might get a response like, well, that's nice, an expression of the solution, but what's the actual value? Meanwhile, an Islamic mathematician might say, OK, then draw me a line with length square root 3. And so here's the important idea. x equals square root 3 is only a solution because we agree that it's a solution, and not because it is in any way better than what Chinese or Islamic mathematicians would have found. Now the reason for that goes back to the Middle Ages, and roughly speaking, since the Middle Ages, European mathematicians increasingly focused on what we would call algebraic solutions. We say that the solution is algebraic if it can be expressed using the basic operations of arithmetic, addition, subtraction, multiplication, division, and roots, applied a finite number of times. And so this expression, which only involves roots, addition, subtraction, is an algebraic solution. On the other hand, if we write something like this, it's not algebraic, because the ellipses here indicates that we're going to perform an infinite number of additional operations. And this is why the Chinese solutions are not considered algebraic solutions. And we might go a little bit further. x equals arctangent of 3 or x equals log 5 are not algebraic solutions. They don't use the basic operations of arithmetic applied a finite number of times. Now since the distinguishing feature of an algebraic solution is the nth root, we often say that algebraic solutions are solutions in radicals. So part of what makes the story of the cubic very interesting is that bad luck seems to have followed everyone who tried to solve it. If we wish to engage in a little melodrama, there is a curse of the cubic. Our story begins in Renaissance Italy, and so in fair Bologna do we set our scene. By 1500 the Italian peninsula became the mathematical center of Europe and the University of Bologna the intellectual center of Italy. Luca Paccioli was among the faculty and Paccioli attempted to find an algebraic solution to the cubic but failed. Consequently in his Suma, a compendium of mathematical, geometrical and artistic knowledge, he declared that the cubic was impossible to solve. We might regard Paccioli's comment as a way to salv his own ego because Scipione del Ferro also taught at the University of Bologna around the same time and he disagreed with his colleague. And around 1501 he found a solution to what was then called the Cosa and Cube equation. Now an Italian Cosa is big and it's a way that the Italian algebras refer to the unknown. So these are equations we'd write as x cubed plus px equals q, where p and q are positive constants. And here's our Cosa, some number of things, and cube. Now an important factor in the story is the nature of academic jobs in the 16th century. In mathematics your reputation was based on two things. Your ability to solve problems, that's kind of obvious, but your ability to pose unsolvable problems that you could still solve. This meant that if you could do something no one else could do, you were assured of continual employment. And whatever you did, you didn't make your method public. Now eventually del Ferro retired and taught his method to one of his favorite students, Antonio Maria Fiori, who lived around 1555 from Venice. And so now a little bit of history. In 1512 French forces invaded Italy as a result of a series of dynastic wars that are kind of complicated to explain and might make for a good series of novels, maybe set in some sort of fantasy setting with dragons. Where was I? Oh yeah, the French invasion went through Russia. Because the inhabitants of Russia fought back a little too effectively, the French retaliated and massacred thousands of inhabitants of the city. One of the survivors was a 13-year-old boy named Nicolo Fontana. However a French knight, the Paragon of Chivalry, slashed Fontana across the face with a sword which left the boy with a permanent speech impediment. And subsequently Fontana became known as Tartalia, the stutterer. In 1534 Tartalia moved to Venice. At the time Venice had a reputation as a town with wild raucous parties. And not much has changed. Now remember Fiori? Fiori had established a reputation as a mathematician while Tartalia was a boy from the backwoods. For reasons that are unclear, Fiori, the established and well-known mathematician, challenged Tartalia, the nobody from nowhere, to a duel. An academic duel. Each would pose 30 questions, and the loser would pay for 30 banquets. Tartalia posed a variety of problems, but Fiori only posed one type of problem. He posed 30 closer and cube equations. When Tartalia saw the questions he panicked. He had no idea how to solve the closer and cube equations. At least not at first. Of February 12-13, 1535, a few days before the answers were due, he had a sudden inspiration. And with that inspiration he solved all 30 of Fiori's problems. And established his reputation as one of the preeminent mathematicians of northeastern Italy. And for now we'll leave Tartalia after his stunning victory and go west to the Duchy of Milan. And here we'll pick up part two of the story of the Curse of the Cubic.