One of the contributing factors they mention in the introduction is that long-distance running (more than 5K or 3.6M) had become a part of the hominid hunting technique.
While there is some support for this position, there is equal, if not more, support for the notion that a higher protein diet is directly responsible for human brain growth during this evolutionary period.
Our ancestors, beginning with H. erectus, shifted to a hunting and gathering lifestyle that required higher levels of aerobic activity [21–24], with morphological evidence showing adaptations for increased long-distance trekking and the adoption of endurance running (ER; aerobic running for distances of more than 5 km) as a new hunting method [17,18].
Importantly, the human brain (adult) uses about 25% of total calorie intake just for brain function (this number approaches 60% in newborns, which is why breast feeding [milk protein and lipids] is so important). In order to support that much energy consumption (consider that the average ape brain uses only 8% of its caloric intake for brain function), proto-humans needed to change their diets from mostly plant material to the higher calorie animal meat and bone marrow. This evolutionary step likely allowed proto-humans to maintain smaller and shorter digestive systems than more plant based apes.
According to Katharine Milton [The Critical Role Played by Animal Source Foods in Human (Homo) Evolution, 2003]:
Without routine access to ASF, it is highly unlikely that evolving humans could have achieved their unusually large and complex brain while simultaneously continuing their evolutionary trajectory as large, active and highly social primates. As human evolution progressed, young children in particular, with their rapidly expanding large brain and high metabolic and nutritional demands relative to adults would have benefited from volumetrically concentrated, high quality foods such as meat.Richard Wrangham has added to this idea that cooking meat (and other foods) made many foods more bioavailable, which allowed hominids to digest more calories from the foods they consumed, again providing much needed energy to maintain the larger brains we were growing.
What spurred this dramatic growth in the H. erectus skull? Meat, according to a long-standing body of evidence. The first stone tools appear at Gona in Ethiopia about 2.7 million years ago, along with evidence that hominids were using them to butcher scavenged carasses and extract marrow from bones. But big changes don’t appear in human anatomy until more than 1 million years later, when a 1.6-million-year-old skull of H. erectus shows it was twice the size of an australopithecine’s skull, says paleoanthropologist Alan Walker of Pennsylvania State University in State College. At about that time, archaeological sites show that H. erectus was moving carcasses to campsites for further butchering and sharing; its teeth, jaws, and guts all got smaller. The traditional explanation is that H. erectus was a better hunter and scavenger and ate more raw meat than its small-brained ancestors. (Ann Gibbons, Food for Thought, Science Magazine, June 15, 2007)Finally, Mary Nassar and her team (Language Skills and Intelligence Quotient of Protein Energy Malnutrition Survivors, 2011) looked at the effects of protein energy malnutrition (PEM is the most common and the most debilitating form of malnutrition) on the physical and cognitive ability of children:
The study was conducted on 33 children aged 3–6 years who suffered from protein energy malnutrition (PEM) during infancy in comparison to 30 matching children to assess the long-term deficits in cognition and language skills. The patients’ files were revised to record their admission and follow-up data and history, clinical examination, intelligence quotient and language assessment were done. The study revealed that 2–5 years from the acute attack the PEM patients were still shorter than the controls and their cognitive abilities were poorer.All of this is to say that there are other and better arguments for increased brain size in human evolution. More importantly, the likelihood is that several factors - increased aerobic exercise, increased protein intake, decreased digestive system - all contributed to the increase in brain size.
Raichlen DA, Polk JD. (2013). Linking brains and brawn: Exercise and the evolution of human neurobiology. Proc R Soc B, 280: 20122250. doi: 10.1098/rspb.2012.2250
David A. Raichlen  and John D. Polk [2,3]Here is the beginning of the article, which lays out some compelling evidence for why they have developed this model.
1. School of Anthropology, University of Arizona, Tucson, AZ 85721The hunting and gathering lifestyle adopted by human ancestors around 2 Ma [editor's note: Ma = millions of years before the present] required a large increase in aerobic activity. High levels of physical activity altered the shape of the human body, enabling access to new food resources (e.g. animal protein) in a changing environment. Recent experimental work provides strong evidence that both acute bouts of exercise and long-term exercise training increase the size of brain components and improve cognitive performance in humans and other taxa. However, to date, researchers have not explored the possibility that the increases in aerobic capacity and physical activity that occurred during human evolution directly influenced the human brain. Here, we hypothesize that proximate mechanisms linking physical activity and neurobiology in living species may help to explain changes in brain size and cognitive function during human evolution. We review evidence that selection acting on endurance increased baseline neurotrophin and growth factor signalling (compounds responsible for both brain growth and for metabolic regulation during exercise) in some mammals, which in turn led to increased overall brain growth and development. This hypothesis suggests that a significant portion of human neurobiology evolved due to selection acting on features unrelated to cognitive performance.
2. Department of Anthropology, University of Illinois Urbana–Champaign, Urbana, IL 61801
3. Department of Surgery, University of Illinois Urbana–Champaign, Urbana, IL 61801
A wealth of recent studies detail connections between physical activity and neurobiology [1,2]. In particular, aerobic physical activity (APA) generates, and protects new neurons, increases the volume of brain structures and improves cognition in humans and other mammals [2–8]. These neurobiological effects accrue during an individual’s lifetime, and a great deal of research has begun to explore the implications of APA for cognitive health . However, recent data also suggest that there is an evolutionary relationship between APA and the brain, including a positive correlation between aerobic capacity and brain size across a wide range of mammals . Here, we review this growing body of evidence suggesting that the relationship between APA and neurobiology exists across evolutionary timescales, and that selection acting on endurance capacity in mammals may have had important effects on the evolution of brain size in these taxa.
In addition to neurobiological effects on mammals in general, this recent work has profound implications for human brain evolution. The human brain is approximately three times larger than expected for our body size, due to increases in several brain components, including the frontal lobe, temporal lobe and cerebellum [9,10]. This major increase in both absolute brain size and brain size relative to body mass occurred during the early evolution of the genus Homo, becoming especially pronounced during the evolution of Homo erectus [9,11–13] (figure 1). Because brain size changes in human evolution are often interpreted in the context of cognition , previous hypotheses for increased brain size in hominins have focused on greater social complexity  or enhanced ecological demands on cognition [15,16]. However, at the same time as brain size began to increase in the human lineage, aerobic activity levels appear to have changed dramatically [17–20]. Our ancestors, beginning with H. erectus, shifted to a hunting and gathering lifestyle that required higher levels of aerobic activity [21–24], with morphological evidence showing adaptations for increased long-distance trekking and the adoption of endurance running (ER; aerobic running for distances of more than 5 km) as a new hunting method [17,18]. Thus, in addition to reviewing patterns of brain evolution in non-human mammals, we propose the novel hypothesis that selection acting
on human locomotor endurance had a measurable effect on the evolution of human brain structure and cognition.
To explore hypotheses linking physical activity and brain evolution, we begin by reviewing proximate mechanisms that allow APA to alter the adult mammalian brain. We then examine intra- and interspecific studies (as well as artificial selection experiments) that suggest selection acting to improve endurance capacity alters these proximate mechanisms and, in the end, affects neurobiological evolution in mammals that have an evolutionary history of endurance activity (athletic species). Finally, we explore correlations between APA and neurobiology across evolutionary time-scales in the human lineage. The purpose of this review is not to suggest that aerobic activity alone is responsible for all aspects of human brain size or cognitive evolution. However, our review suggests that aerobic activity represents a previously unrecognized factor in mammalian neurobiological evolution, and highlights the possibility that noncognitive selection pressures may have played an important role in the development of the human brain.
2. Effects of aerobic physical activity on the brain: proximate mechanisms
Many studies suggest that APA leads to the formation of new neurons (neurogenesis) in some portions of the adult brain [2–4,7,25–27]. In rodents, voluntary wheel-running produces a three- to fourfold increase in neurons in the dentate gyrus of the hippocampus [2,8]. There is also some limited evidence that neurobiological changes associated with APA occur in other brain regions . For example, there is a trend towards increased neurogenesis, and evidence of gliogenesis (generation of new glia that support neuronal activity) with APA in the frontal cortex of rats [28,29], and neurogenerative activity induced by APA was found in superficial cortical layers and in the motor cortex of rodents .
Activity-induced neurogenesis has a major impact on cognitive function and on the size of brain components. For example, performance in memory and spatial learning tasks improves following APA in non-human taxa such as monkeys  and rodents [8,31,32]. In humans, aerobic fitness is positively correlated with hippocampal and basal ganglia volume in children and older adults [25,33,34], with grey matter density in the insula of young adults , as well as with the amount of grey and white matter in the frontal lobe and other brain areas of older adults . These structural changes across many brain regions appear to have significant functional effects. In school-aged children, fitness levels and participation in higher amounts of physical activity are correlated with improved cognitive function [5,26,36]. In healthy young adults (approx. 22 years of age), both acute and long-term APA improves performance on memory tasks, suggesting enhanced hippocampal function . Finally, several studies have linked APA with either improved cognitive performance (especially executive functions and spatial memory) or a reduction in cognitive decline in older populations [3,38,39]
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